ye Theory of the Motion of the Moon; containing a New Calculation of the Expres- sions for the Coordinates of the Moon in terms of the Time. By Ernest W. Bkown, M.A., Sc.D. PART I. CHAPTERS I. -IV INTRODUCTION. The formation of numerical expressions, deduced as a consequence of the Newtonian laws of motion and gravitation, which shall represent the position of the Moon at any time, may be roughly divided into three stages. As a first step we consider each of the three bodies — the Sun, the Earth, and the Moon — as a sphere of mass equal to its actual mass and arranged in concentric spherical layers of equal density. The Earth (or the centre of mass of the Earth and Moon) is supposed to move round the Sun in a certain ideal elliptic orbit, and all disturbances of this orbit and of the Moon from any other source than the ideal Sun and Earth are neglected. This first stage constitutes nearly the whole of the labour of solving the problem of three bodies as far as the particular configuration of the Sun-Earth-Moon system is concerned. When this is done we proceed to the second step, which involves the determination of the effects due to the difference between the actual and ideal motions of the Earth and Sun, to the influences exerted by the other bodies of the solar system, and to the differences between the real and ideal arrangements of the masses of the bodies. The calculations so far may, theoretically at least, be made without any knowledge of the configuration of the system at any given mm 40 Mr. Ernest "W. Brown, Theory of the time or times beyond a general idea of the order of magnitude of certain of the constants involved. The third and final stage consists in a determination by observation of the various constants which have entered into the theory, and the substitution of their values so as to obtain numerical expressions for the coordinates in terms of the time. In actual practice these lines of division cannot be satisfactorily kept, partly from the length and complexity of the calculations necessary to obtain algebraical expressions sufficiently complete, and partly from a similar diffi- culty in comparing large numbers of observations with the results of the theory for the purpose of obtaining the values of the constants involved. Certain of the latter, however, particularly those most frequently occurring, namely, the mean motions of the Sun (or Earth) and Moon, can be obtained, , with very little knowledge of the theory, with sufficient accuracy to enable us to use their numerical values at the outset, and so to save a large part of the labour. The first and second steps may be separated without much difficulty, and it is the first which forms the chief object of this Memoir. Of the methods which have been devised to solve the first part of the problem, those of Hansen and Delaunay must take the first place, not because they were intrinsically the best adapted to the purpose, but because in the hands of their authors they were actually carried out with a degree of detail greater than any other. It is not necessary to discuss here their respective merits ; it is sufficient to recall the fact that Hansen's was entirely numerical, and was made the basis of the tables of the Moon used at the present time, although the latter were published forty years ago, while Delaunay's was, owing to the method used, entirely algebraical. Further, Hansen embraced the whole problem in his theory ; Delaunay only lived long enough to complete the first step ; his work, however, has been carried on by other writers, and will attain its completion in the tables now in process of formation. The earlier method of Laplace, adopted also by Damoiseau and Plana, and that of Lubbock and de Pontecoulant, although perhaps unsuitable to obtain the accuracy required at the present day, attained results which were at the time as much in advance of those previously published as those of Hansen and Delaunay were over all earlier theories. But if we look only to suggestions of methods of treatment, quite apart from the extent to which they were actually carried out, the most fruitful Motion of the Moon. 4i contributions to the first part of the problem were undoubtedly those of Euler, who preceded all the writers just mentioned. His two treatises of 1753 and 1772 contain three distinct methods ; and it is not a little remark- able that the theories of Hansen and Delaunay may be said to be ultimately based on two of them, while the third forms the foundation of the method developed below. Amongst the points of correspondence between the first theory of Euler and the theory of Hansen, it is sufficient to note the manner in which the true longitude and the radius vector of the Moon are expressed. Hansen, however, covered up most of the traces of any such connection by his peculiar method of using a so-called " variable time " and by developing his formula} with the aid of the method of variation of arbitrary constants. His formula} can be otherwise obtained, as Hansen himself and others have shown ; even if it were not so, we owe the application of the arbitrary constants method to celestial mechanics in the first place to Euler. This latter method, which is contained in the appendix to Euler's volume of 1753, was rendered a practical one under the masterly treatment of Delaunay ; but it is doubtful if anyone but the originator of the method would have had the courage to undertake the laborious calculations necessary to bring the work- to a successful conclusion. The theory of Euler, published in 1772, is of particular interest here, since it suggested the method used below. It is based on the use of rect- angular axes, of which two move in their own plane with constant angular velocity, and on a division of the inequalities into classes according to the composition of their coefficients. This work of Euler seems to have received but little attention, apart from the practical results obtained ; no attempt was made to develop his method further. It was reserved for Dr. Gr. W. Hill, over a hundred years later, to take the next step by so altering the forms of the equations, while preserving the original ideas on which they were constructed, that they might be made available for accurate calculation. His two papers, " Researches in the Lunar Theory," published in the first volume of the American Journal of Mathematics, and " On the Part of the Motion of the Lunar Perigee which is a Function of the Mean Motions of the Sun and Moon," published separately, and also in the eighth volume of the Acta Mathematica, showed what the method was capable of effecting, and opened out a new region for practical calculations and theoretical researches. Royal Astron. Soc, Vol. LIII. h 4 2 Mr. Ernest W. Brown, Theory of the " Dans cetts oeuvre," says M. Poist:are in the preface to the first volume of his Mecanique Celeste, " il est parmis d'apercevoir le germe de la plupart des progres que la science a faits depuis." In this connection the work of Adams must be mentioned. Starting from an entirely different point of view, that of effecting an accurate determination of the mean motions of the principal arguments, he obtained the corresponding part of the motion of the node by a method somewhat similar to that used by Dr. Hill for the perigee. One of the main difficulties in the latter case — the reduction of the equations to a suitable form— does not occur in the case of the node, and there is no indication in the single paper which Adams published on that subject in the Monthly Notices for 1877 that he contemplated the use of moving rectangular coordinates. But that he was investigating the properties of the latter may be inferred from another paper on the lunar theory in the same volume of the Monthly Notices. During the last six years I have been attempting to develop the ideas contained in the " Researches " by calculating the coefficients of terms with certain definite characteristics.* Dr. Hill had obtained those which had the characteristic unity, that is, which were functions of the mean motions of the Sun and Moon only, and also that part of the motion of the perigee which was a function of the same quantities ; Adams had done the same thing for the motion of the node. It therefore remained to obtain the general equa- tions, to put them into forms suitable for calculation, and to show how the other parts of the motions of the perigee and node might be obtained. Experiments were made with the inequalities whose characteristics are the first, second, and third powers of the ratio of the mean parallaxes of the Sun and Moon, and the same powers of the eccentricity of the Moon. The forms of the equations which were there used were, however, troublesome, chiefly from their liability to produce errors in the actual calculations. In a paper which will be hereafter referred to, for the sake of brevity, as the <! Investigations," f I showed how this difficulty might be avoided without, I * The " characteristic " of any part of a coefficient is defined to be that part in its expression which consists of powers and products of the eccentricities, the inclination, and the ratio of the mean parallaxes. The other factor is a function of the ratio of the mean motions, and it also depends on the nature of the coordinate used. See Chap. I. §15, below. t "Investigations in the Lunar Theory," Amer. Jour. Math. vol. xvii. pp. 318-358. Riff '.lit Motion of the Moon. 43 think, causing any increase in the labour of making the calculations, and it has certainly diminished the actual time required for their performance. In fact, at least three fourths of the calculations might be performed by a com- puter whose stock-in-trade amounted to little more than a thorough know- ledge of logarithms. An effective control over the computations can be kept at almost every step ; and as the operations which would be turned over to the computer are always the same, he would soon be able to do his work with very little supervision. It is intended to apply the method so as to completely solve the problem of the Moon's motion as far as it is affected by the Sun and Earth alone, the action of the" Moon only on these bodies being included, and the three bodies being treated as particles of equal masses. The degree of accuracy aimed at is that the coefficients of all periodic terms in longitude, latitude, and parallax shall be included which are greater than o"'oi, and that they shall be correct to this amount. The number of terms required is undoubtedly very great. The calculation of coefficients up to the sixth order inclusive with respect to e, k — the lunar eccentricity and inclination — will be necessary ; those of the seventh order may be replaced by their elliptic values. The corresponding orders for ef, a — the solar eccentricity and the ratio of the mean parallaxes — are deduced from the fact that c /2 is roughly of the order e 3 or k 3 , and that a is of the order e 2 or k 2 . To obtain the coefficients with the above-mentioned degree of accuracy it will be necessary to calculate terms contained in about one hundred characteristics. These will include about five hundred periodic terms, and will require the actual calculation of perhaps two thousand separate co- efficients. The results now published contain the terms present in fourteen characteristics — that is to say, about one-seventh of the whole. Notwith- standing the fact that the number of terms contained in the higher charac- teristics is much greater than that in the lower ones, the work done so far probably amounts to more than one-fifth of the whole. This is due to the fact that a much higher degree of accuracy is required for the lower terms than is actually necessary to obtain the corresponding coefficients correctly to o" - oi ; the presence of small divisors causes a loss of accuracy, which has to be continually borne in mind in judging of the number of places of decimals which are to be calculated. For example, the term, the mean motion of 44 Mr. Eknkst W. Blown, Theory of the whose argument is twice trie difference between the mean motions of the perigee and node, requires that the calculations be actually carried three places of decimals further than would be necessary for a term of the same order with no small divisor. Fortunately, however, the majority of the terms which cause the most trouble, due to the presence of small divisors, are those which contain both ef, a in their characteristics, and therefore the number of characteristics of this nature to be considered is much smaller than would otherwise be the case. The theory will be an algebraical one throughout, with the single and important exception that the numerical value of the ratio of the mean motions of the Sun and Moon is substituted. The reasons for this may be briefly stated as follows : — First, slow convergence of the series which represent the coefficients arranged according to powers of m or m* e, k, e', a, takes place mainly along powers of m ; secondly, the value of m is known from observa- tion with great accuracy ; thirdly, estimates would have to be made of the values of the constants e, k used in this theory from the values of differently defined constants of eccentricity and latitude used in other theories ; fourthly, wery little, if any, extra trouble is caused by leaving e, k, e', a arbitrary. Thus the theory, while remaining to a large extent algebraical, will possess all the advantages of a purely numerical theory. It may be also mentioned that, by combining the results of this theory with Dr. Hill's modification of Delaunay, it can be effectively used for researches by the method of the variation of arbitrary constants. The procedure is. intrinsically contained in my paper " On the Theoretical Values of the Secular Acceleration in the Lunar Theory " in the Monthly Notices for March 1897. Of the four chapters which are now published, the first contains the whole theory, with certain exceptions, so far as it is necessary for the con- tinuous development of the numerical results. The exceptions are those parts of the theory which refer to numerical results previously obtained and which are not necessary for those which follow. For example, all details of purely theoretical interest are omitted, and no account is given of Dr. Hill's method for the determination of the intermediate orbit used here (which I • As usual m—n'ln, m=n'l(n-n'), where n, n' are the observed mean motions of the Moon and Sun. Motion of the Moon. 45 have called the "variation curve"), or of those methods used by him and Adams for finding the principal parts of the motions of the perigee and node. All that is necessary is a quotation of the numerical results, and they will be found in their proper places with the sources from which they have been obtained. The following is the table of contents :- Chapter I. — General Development of the Theory. Section (i). An investigation of the disturbing function used, with the necessary corrections. Section (ii). The two forms of the equations of motion. Section (iii). Development of the disturbing function according to powers of i /a, z, e'. Section (iv). The form of the solution. The general system of notation adopted to represent the coefficients, arguments, &c. Section (v). Method of solution. Preparation of the equations of motion. Section (vi). Exact definitions of the arbitrary constants used in the theory. Section (vii). Methods used for the solution of the equations of condi- tion satisfied by the coefficients. The long and short period terms which give rise to small divisors. Manner of obtaining the higher parts of the motions of the perigee and node. Section (viii). Details concerning the numerical calculations and the methods used to verify them. Section (ix). Transformation to polar coordinates. Chapter II. — Terms of zero order* Numerical results. Chapter III. — Numerical results for terms of the first order. Chapter IV. — Numerical results for terms of the second order. Future chapters will contain the terms of the third, fourth . . . orders. With regard to the calculations, no trouble has been spared to secure correctness. Errors are of two kinds, those which are merely numerical and those which are partly algebraical — i.e. due to the use of a wrong series of factors or to the omission of some series of terms. To test the former, equations of verification were computed at every step, and the nature of the method rendered these very numerous. An error of the latter kind may * The word " order" here and elsewhere refers only to e, e', k, a, and not to m. 46 Mr. Ernest W. Brown, Theory of the escape such, a verification, and will generally produce a large discordance ; the results were therefore tested by a rough comparison with those of another theory — say, that of Delaunay. One searching test of this kind has been applied to the majority of the terms now published, namely, the compa- rison of the motions of the perigee and node deduced therefrom with those deduced by Hansen and Delaunay. The results obtained so far point to certain appreciable errors in the theories of Hansen and Delaunay. Dr. Hill has shown * that the last two terms of Delaunay's expression for the part of the motion of the perigee which depends on m only are wrong. The part of the motion of the perigee which depends on e 2 I have calculated in two quite different ways, and it appears that the last two, if not the last three, of these terms of Delaunay's expression are seriously erroneous.! An error in Delaunay's expression for the part of the motion of the node which depends on e' 3 was actually traced down to an error of transcription in his theory 4 Hansen's theoretical value for the annual motion of the node appears to be at least one if not two seconds in error.§ Professor !Newcomb in his discussion of the results obtained for the coefficient of parallactic inequality || considers that Hansen's value is about o"'30 in error : this amount, though small in itself, is of importance if the coefficient be used to obtain the solar parallax. I hope before long to finish the computation of this coefficient so as to obtain it within o //, 02, and, at a later period, the values of the annual motions of the perigee and node, so far as these depend on solar action only, correct to o"'Oi. CHAPTER I General Development of the Theory. Section (i). — The Problem of Three Bodies and the Disturbing Function. i. The disturbing function used by all investigators except Hansen gives only a portion of the inequalities produced by the Sun in the motion of the Moon. This form of the disturbing function will be used below, and * Annals of Mathematics, vol. ix. pp. 31-41. f Monthly Notices, vol. lvii. p. 336. % Ibid. p. 341. § Ibid. p. 340. I Astron. Jour. vol. xv. p. 167. Motion of the Moon. 47 therefore we shall, in this section, investigate the small additions which must be made to it in order that the whole of the effect of the Sun on the Moon's motion may be obtained. The method of this section is similar to that of Prof. Xewcomb in " The Actions of the Planets on the Moon " (Amer. Eph. Papers, vol. v). Let X, Y, z, r be the coordinates and distance of the Moon referred to axes fixed in direction and passing through the centre of the Earth ; x', y' , z', i J , those of the Sun referred to parallel axes through the centre of mass of the Earth and Moon ; M, E, m' , the masses of the Moon, Earth, and Sun reckoned in astronomical units. The ^-coordinates of the Moon, Earth, and Sun referred to parallel axes through the centre of mass of the three bodies are respectively EX _ m' x' E+M m' + E+M' _ MX _ ml ri_ (E+M) x' E+M m' + E + M' m' + E+M' with similar expressions for the other coordinates. Let T be the kinetic energy of the system relative to the centre of mass. Then 2 t=m\ ( J- - dJC - ' \ \ E+M dt m' + E+ M dt M dX dx \ 2 + "\\ here 7 +e\( m dx +- ?*'_^V+ 1 \E+M dt ^m'+E+M dt) + " - \\m' + E+M dt) ^ • • • I" • • • , + , — EM Pl E+M' /j. 2 = 'm' + E+M' Let F be the potential energy of the system. By Lagrange's equations we then have for the Moon's motion relative to the Earth— ft d*XJdF dt* dX' H-\ cPYjdF "if- ar hi d 2 z = oF dfi dx ' and for the Sun's motion relative to the centre of mass of the Earth and Moon — Hi d-x' dF H-2 dh,' dF ' dt- ex' ' r " z dt- 6y F being expressed in terms of X. Y. z. a/, y' , z 8V dF at 1 o~ 4 8 Mr. Ernest W. Brown, Tlieory of the Let r{ be the distance of the Sun from the Earth, A its distance from the Moon. Then „ EM , Em . Mm' , 1< ■=. 1- — r H — — , r t{ A . . and r 2 =X 2 +F 2 + s 2 : r'^x'z + y't + z'*, where S is therefore the cosine of the angle subtended by A at the centre of mass of the Earth and Moon. 2. We shall first consider the effect of the Moon's motion on the Sun and determine the deviation of the Sun's motion from an elliptic orbit in the plane of reference. For this purpose we may omit the first term of F, since it does not contain x' , y', z' . Then ^ jE+M W a/ Expanding i/r/, i/A in powers of r//, we obtain F i '_L^j.w\r i x EM ri (3&-i\+ _-(m +E+M)^ J + lWTWyi r , 3 [ 2 $ 2 )+ • • • _• The order of the second term in comparison with the first is in the ratio Mr % : Er' 2 =i : 12,000,000 approximately. The order of the next term is in the ratio Mr* : Er' ? >, a quantity which may and will be totally neglected. A sufficient correction to the elliptic motion of the Sun about the centre of mass of the Earth and Moon may therefore be obtained by using (m+E+M) w+MYir .ja- 2 ) the disturbing function, and substituting for the coordinates of the Mcon their elliptic values, together with the principal inequalities due to the Srn, as Motion of the Moon. 49 as found in Chaps. II., III. below. If ri be the observed mean motion of the Sun, we define the mean distance a' by the equation 3. Next, for the motion of the Moon we have, on expanding F and rejecting the useless term m'(E+M)//, /a, r ^ EM \r/ A _E+M , m'r 2 r ^^->e1m^-Is) &-EM+M2 tY3S S 4_ I S S2 + 3A (E+Mf r' 2 U 4 8> + (E+Mf r'\ 8 b 4 " + ¥ b J + • • • J * The expansion is carried as far as will be necessary, since we shall neglect quantities of the orders r 4 // 4 , M> s /E? /3 . The force-function which we shall use is that ordinarily used, namely, r LvV 2 -2rr'S+r 3 r' r' 2 J r r i [_\2 2) v\2 2 J <(^-¥^)+--] - <■> In this, the elliptic values of x', y 1 (Y=o) are substituted and m' is put equal to n n a' s instead of n'' 2 a' 3 -E-M. It is necessary to consider what corrections must be made when Si is used instead of F/ /J , 1 . 4. The corrections are of three kinds and are so small that we need only consider their effects to the first order of the disturbance. (a) Correction due to putting n' 2 a' 3 =m' instead of m' + E+M. We must add to 8 the terms r' 6 \_2 2 r \2 2 /J liOYAL AsTIiON. SOC, VOL. LIII. T 5 o Mr. Ernest W. Brown, Theory of the This correction will be sufficiently accounted for if we multiply all the inequalities due to the Sun by ._ E+M __ _ i 330 000 approximately.* (b) Correction due to using the elliptic instead of the true values of the solar coordinates. We must add to 8 the term »'v{^0s.-I)], where S operates on a/, y', z', and 8/, St/, In! are the perturbations of a-', y', z', as found in § 2. (c) Correction due to the difference between F/^ and Si. Since the 2 ., 3 terms in F/^ or S3 involving n'V 8 -^ 1 ^ j give inequalities having the factor U/a') J , where a is the " constant " of the Moon's distance, this correction may be partly made by multiplying the inequalities having the factor (a/a')' D y (E-MV \E+MJ ' To the order considered here, there will then remain to be added to S3 the term ^ ri rn EM »V35 S 4_f5 s4 + 3Y n a —^, 5. "The method of procedure is therefore as follows : — The values of X, Y, z are first obtained by using q with m'=n' 2 a' 3 , z' = o and elliptic values for x', y'. With the values of X, Y, z thus formed we compute lx' , h/ , W by means of the disturbing function in § 2. The corrections to, X, Y, z, noted in (a), (b), (c) above, are then easily obtained. The first step is that * Except for the mean motion of the perigee where the second and succeeding terms, which are of the order of the square and higher powers of the disturbing forces, are a little Greater in actual value than the first term. The correction may be made with sufficient accuracy by multiplying the mean motion of the perigee by 2 to' 2 ' m' 2 m' See Monthly Notices, 1897 June. Motion of the Moon. 5 l which very frequently bears the name of the " Lunar Theory." It will be noticed from this investigation that the elliptic values of the solar coordinates to be used are those referred to the centre of mass of the Earth and Moon. Section (ii). — 77*6 Equations of Motion. 6. Let x, y, ~, be the coordinates of the Moon, referred to rectangular axes through the Earth's centre, of which those of x, y are in the plane of the Sun's orbit (supposed fixed and elliptic), the positive direction of the moving .r-axis being constantly directed to the mean place of the Sun ; r i =x i + y i + z i ; p->-=x? + y* ; n, n! , the observed mean motions of the Moon and Sun ; r', e', a', the radius vector, eccentricity, and semi-axis major of the Sun's orbit (§ 5) ; v, the solar equation of the centre ; Si=aj cos v + y sin v. So that ■rr'8=Xx' + Yy' + zz' =r'(x cos v + y sin v)=r'S 1 , and .■2— n i =p»+.' Also, representing V—i by 1, let u=x + yi, s=x — yi, ns=p- ; ri E + M m= n — n'' (n — n')- £=exp. (n — n')(t — t )i, n 1 d yd (n — n')i dt d'C, where t is a constant to be defined later. The equations of motion, referred to the moving axes and with the force- function a (§ 3), are d'x ,dy , 2 'dp, dt 1 dt ex d 2 y , ,dy „ op df- dt J ay dh _o P dt* ?-~ ' 52 Mr. Ernest W. Brown, Theory of the Let 2 2 and transform to the independent variables u, s, z and the dependent variable t The equations become D 2 w+2m2)«=- DH = - 2 do/ (n — n') 2 ds 2_ 9o_' (re — «') 2 dn ' _i 8 a'. (n—n') 2 3s where, by Sect, (i), ^-{us + z'r L(r'*- 2 r'S 1+ M 8 + ^ r> r"J T 2 This gives, on expansion according to powers of i//, after some trans- formations, (re — re') 2 (us + z-y 4 W here Oi==3 m " + S S 1 3- 3 S 1 («« + ^)" m 2 a' 4r + a r m 2 a^_ a 12 ' r' s L 4 m 2 a' 6 (2) a' 8 ' r' 6 |_ 4 ' 6 _3g 1 5_3S Sia ( MS + s 2 ) + iS Si ( Mg + ^ + suppose, where «, is the part of s? x which involves u, s, z to the degree </ ; Si is of the first degree with respect to u, s. It will be noticed that w 2 is zero when e?=o. <p Motion of the Moon. The equations may now be written— (D + m) 2 u + xn 2 u+^m 2 s — (us+z 2 )i ds ks 3 q , (D— m) 2 s + m 2 s -f 3 m ? u — — v — 2 2 (ms + s 2 )'* clu (Z>»_m*) 5 - ,— -f = -A?> (mS + S 2 )* J OS 53 (3) (3') (4) These are the fundamental equations in the theory. Since u, .<? are conjugate complexes, either of the first two equations is sufficient ; we shall use the first in the calculations. 7. Homogeneous Form of the Equations. — Multiply the three equations by Da, Du, 2l)z respectively, and add. We obtain D\Du . Ds + (Dz) 2 + 3 m 2 (u + s) 2 -m ,i z 2 + — 1 L 4 (us+z 2 yj L ds du dz J Since S3 1 is a function of u, s, z, r', v only, and since ?•', v are supposed to be known functions of the time, du ds dz dt and therefore the right-hand member of the previous equation = d ^-Dt-D &> = .-- ' d ^-D Si} dt (n—n')i at =D' Q^-D&^DiD-WQi)- Si ,]. where D~ l denotes the operation inverse to D [i.e. integration with respect to £ followed by a division by £), and X^'S^ denotes the operation D performed on Si j only so far as £ occurs in r', v. With this form of the right-hand member we can integrate and obtain Du . Ds+(Z)z) 2 + - i m 2 (u + sy-m 2 z 2 + {us + z'f --C'-Si,+D-\D'i (5) Now add this to the sum of (3), (3'), (4) multiplied by .<?, it, 2; respec- tively. Since, by Euxer's theorem, 'IS (« (Z .._.. ' 54 Mr. Ernest W. Brown, Theory of the we obtain an equation which may be written — -- D%<us + z*)-Du . Ds-(Dzf-2m(uDs-sDu) + ^m%u + sf-3rn 1 z a Also multiply (3) by s, (3') by u and subtract ; multiply (3) by z and (4) by u and subtract. The two resulting equations are D{uDs-sDu-2mus)+tm\i?-s i )=s d ^-u d ^ ( 7 ) 2 B(uDz-zDu)-2mzDu-m !> uz-lm i z(u+s)=z ®-k-± u 4jj± - ( 8 ) Instead of the last we may obtain the more symmetrical form, D[(u - s)Dz - zD(u — s)] — 2mzD(u + s) - m 2 «(w - s) \ os ou J 2 oz (8') The equations (6), (7), (8) will be called the homogeneous equations. Their left-hand members are homogeneous and of the second degree with respect to u, s, z, while their right-hand members are, abstraction being made of C, of the 2nd, 3rd . . . degrees with respect to the same variables.* Section (iii.). — Development of S a . 8. In the last section, the development of S?j according to powers oil /a' has been given ; the result is there numbered (2). We must now further develop it according to powers of e' and z, which are small quantities of the first order. The development will not be carried beyond quantities of the orders — a 3 a 2 , a 2 „ 2 a , 3 ,., „'3' „<a e ' "di" ' a' ' ' G, " Or & Lb By the definition of Sj we have S { =x cos v + y sin v =~~ (ue~" +se v ') 2 * Fuller explanations of the transformations in the section are given in the Treatise cm the Lunar Theory, Chap. II. (iii.). Motion of the Moon. 55 Substituting in (2) and remembering that Si 1 =<^2 + °>3 + w 4 + <°S> we have, to the orders of small quantities just mentioned, 3 .4 (w 2 a 2 4-s 2 a 2 ) + -t«B 2 — « 2 b 2 L w 3=- 7 u .3/,/ 2 (w 3 a 3 + s 3 a 3 ) + j?(w 8 sc 3 + ush 3 )—^uz% - J ss "c «L 2 f35 4 a' 2 L64 "4—^1 >-■ (** 4 fl4 + * 4 a 4 ) + 5 , (m 3 «c 4 + ms 3 c 4 ) + - J w 2 s 2 f 4 ~ 2 ,64 16 3 2 S 2 f -„ S U% + * S « 2 C 4 + 9 M6'[' 4 j where - 6 -o (W 5 + « 5 ) + - 3S „ (« 4 « + MS 4 ) 4- J 5 /«¥ + W*S 3 ) ,128 128 64 2 .,. 3 (Ji=- ■--. « '"" r " t 1 2= ,- 3 -r> tl 4=;,,- ) a c -i= .-, ( r and a 2 , f) 3 , . . are the values of n 2 , h 2 , . . when — 1 has been put for 1. 9. The quantities a 2 , i 2 , are to be expanded in powers of e' ; they are well-known elliptic expansions and they have been computed by several investigators. They may be conveniently obtained from the expressions given by Delaunay in chap. ii. vol. i. of his Theorie de la Lune by giving to the angle, there called a, suitable values, or from the tables of Cayley {Mem. R.A.S. vol. xxix., Coll. Works, vol. in). Putting Exp. l'i=£ m (see § n, below), where I' is the solar mean anomaly, we obtain n 2 = = _5 e /a + i3 e /4 2 16 + . £ 2m + _ r _ e '4£4m 24 1280 384 V2 10 128 J I 7g'2_ II 5 e '4V^2m 2 " 6 J + l 4 8 768 J C 533 16 e' 4 C" 228347^ 3840 56 Mr. Ernest W. Brown, Theory of the 2 2 8 \2 16 128 ;\. J 4. .e' 3 C* m 4. l6 V 3 £~ 3m , 4 4 lr 6 '2^2m + 53 e '2^-2m C :< =i + 2 3g'3^3m 4. 7 7 e '3^-3m j 12 6 a 4 =i- 3 e'£ m + I 3 e'£- ra 2 2 The values of H„ c„ . . . are obtained from those of a. 2 , c„ . . . by putting i/£for£. Section (iv). — Form of the Solution. Notation. 10. Let Fbe the true longitude in the plane of X Y or x y reckoned from the fixed axis of X and 4, the latitude above this plane. ^ Also let 1>, Z, f , F denote the same angles as in Delaunay's theory, Chap. xi. vol. ii. i.e. let D=(«-w')« + c-e'=Half arg. of the "Variation," I =cnt + e-m =Arg. of the Principal Elliptic Term, Z' =«'< + £' -V = „ „ " Annual Equation," 1?—g n l + t -.$ = „ „ Principal term in Latitude. Motion of the Moon. 57 Here, as usual, e, sr, 6 are the values of the mean longitudes of the M and of its perigee and node at time t=o ■ e' is the mean longitude of the at time t=o and to' the (constant) longitude of its perigee; (r- ( i -g)n are the mean motions of the perigee and node. We then have X= P cos V, Y=p sin V, z=p tan $=r sin <j>. Whence, as we shall put £ = exp. Di, (§ n), x—pcos(V-n't — e')=pcos(V-nt — e + D), 2/=psin(F-n7— e')=p sin (T-««-e + D) ; oon :un w =pexp. (V—nt — £ + D)i utr l =p exp. (F— m« — c) (, s =pexp. -(F-w<-e + D) t s£=pexp. — (V— nt — e) t (9) It is well known that, with the limitations here imposed, r, V- n?-e, <j> are expressible by sums of periodic terms whose arguments are (algebraic)' sums of multiples of the four angles 1), I. /', F. Hence ,i-, y, z are Expressible in the forms X\ COS") y[=&%A i:F:Clir sin' (iB+pl + rl 1 + qF), h P, q, r=o, where a is a linear constant and A a coefficient ; the sign of summation 2 denotes that the sum of all such terms must be taken. ° Remembering the definition of u, *, it is evident that u, s, a may be therefore expressed in the form za ^ A i,p,<!,r exp. (iD+2)l + rl' + qF) with certain limitations which will be set down later. There are four sets of notations required. First, for the exponentials : secondly, for the constants of distance, eccentricity, latitude and ratio of the mean parallaxes ; thirdly, for the numerical parts of the coefficients (the>o parts are functions of m only, and the numerical value of m is used thrcno-h- out) : fourthly, for the terms of u, .*, z which are of a particular order. Royal Astrox. Soc., Vol. LTif. ,. 5 8 Mr. Ernest W. Brown, Theory of the ii. Notation for the Exponentials.— Recalling that m= ft' / (n~-n'),^emay evidently write B = (n-n') (t~t ), l=c (n—n 1 ) (t—t x ), l'=m (n-n')(t-t 3 ), F=g(n-n')(t-t i ), where c (w— n')=cn, g (n—n')=gn, and the signification of t Q , t 1} t 2 , t 3 is obvious. We have, in Section (ii), defined K by the equation £=exp. (n—n') (t — t e ) i, and we now give to t in this expression the same meaning as it has in D, so that £=exp. Di. Let, for a moment, £ c =exp. c{n-n')(t-t{)i.. Remembering the definition of the operator D we have where i, j, p are positive or negative integers. Now. in the method pursued here, we shall always proceed by equating to zero the coefficients of like powers of K, K° c in equations which consist of such expressions as that just written down, and it will never be necessary to substitute the value of c in the indices ; its value is only substituted in the coefficients. The above equation shows that the coefficients will be the same whether we write K%\ or, f s \ Further, as the suffix c always occurs in the index whenever it is present as a suffix, the suffix is unnecessary for purposes of distinction and we shall omit it in future. The same remarks apply to / m =exp.»A £' ig =exp.5Fi. AVe may therefore put exp. (tD + P l + rl' + q-Fy^"*™***, the index of K always denoting the coefficient of t in the corresponding argument divided by n~ri. p p. Motion of the Moon. 59 12. Notation for the Arbitrary Constants and the Parameters. — There are six arbitrary constants present in the solution. Three of these— e, ct, 0, or f o, t ly 4 — have already been considered ; they are contained in D, I, F, or in £, £ c , K g - The other arbitraries to be used will be denoted by n, a, e, k, connected by one relation. The exact definitions of c, k,— the constants of eccentricity and inclination— will be found in Section (vi) ; n has been defined as the observed mean motion ; the linear constant, a, is connected with n, e, k U+Mbj a relation which will be defined in the same Section. In elliptic motion, this relation would be n 2 a:'=E+M; in the actual case the relation differs a little from this. The parameters in powers of which expansion will be made are m, e, e', k, a= a a' The numerical value of m is used, but the other parameters are left arbitrarv. 13. Notation for the Numerical Coefficients. —From what precedes, it is evident that u, s, zi may be expressed in the form =a2.4. ,rV + * or, as it will be more convenient to write it, >P, q, r= + co The latter form has the following properties, which are easily deduced from the known properties of the expressions for V, r, <£. (a) sK is deduced from nXr x by putting i/f for % ; if i/Cbeput for ? in the expression for zt, the coefficient merely changes sign ; (b) i is odd or even according as A contains odd or even powers of a ; (0) g is even in the expressions for u%-\ hK and odd in that for zi ; (d) A is of the order e w e' 1 " k 1 *' at least, and it contains higher powers of e, </, k which differ from \p\, \r\ , \q\ by even integers. 14. I now give the general notation adopted throughout. It is devised so as to represent every part of every coefficient. 65 Mr. Ernest W. Brown, Tlieory of the The general term in uf 1 or z t will be expressed by a(€ p+ »'E' p V +r V r '^ +s '/fc'' J 'a*') i e J,+2p 'e' r+2 '''k s+2? 'a , 'C 2i;fc|,cira±M ( I0 ) where * p, q, r, /, q', r', s'=o, i, 2, ... , 2i=o, ±i, ±2, ... The coefficient ( ) t above written corresponds to the upper signs in the index of ?• With the lower sign ofpc, interchange the indices of e, e' ; With the lower sign of <rm, interchange the indices of v, V ; With the lower sign of qg, interchange the indices of l,li. The sum of all such terms for all values of p, q, r, p', q', /, «', 2% gives the complete expression of uK" x or of zi. From the properties (a), (b), (c), (d) just given, it is evident that when q is even, (io) gives the general term of uK" x ; that when q is odd (io) gives the general term of zt, and that then that *r is deduced from uK~ x by putting i/Kfor ?; that J, zi are odd or even together. When 2i is odd, we shall frequently denote this fact by putting 22 = 2^, so that h will denote an odd multiple of ±-|. When an index of any symbol inside' (), is zero, the symbol is simply omitted. In the cases of the coefficients of the first order, namely, (e)„ (e% (?)<> (v')i, (*)<> ( £ 0» tne brackets ( ) will be omitted for the sake of brevity, as they are unnecessary. Particular Case.— In the case where p, q, . . . s' are all zero no letter would occur inside ( ),. This being inconvenient we shall denote the corre- sponding coefficient by a,. Thus, the terms independent of e, d k, a in uK~ x are denoted, by nSfiff 1 , i=o, ±i, ±2, . . . These are the terms of order zero (§ 15 below). There are no such terms in z. * No confusion will be caused by this new use of the letters r, r', since they only occur, in this sense, in the indices of rj, t/, «', and have positive integral values. Motion of the Moon. 15. Characteristic and Order. — The factor 6x of the general coefficient will be called the characteristic part of the coefficient or, briefly, the characteristic. The Order is the sum of the indices of e, e 1 , k, a. The order of the general term is thei-efore p f 2p' + r + 2r' + q + 2q' + s'. The word " order," as used here, is thus independent of m — a necessary restriction, since the numerical value of m is substituted at the outset. A few remarks and an example may make the notation laid down in § 14 clearer. It will be observed that e, e' are always associated with e ; rj, ■>]' are always associated with e! ; k, k' are always associated with k ; a is always associated with a=a/»' ; while, as is well known, e is associated with the index c ; e' with the index m ; k with the index g ; and an odd power of a with an odd value of 21. If the numerical value of m had not been substituted at the outset, we could further have denoted the particular power of m involved bv inserting m J inside ( ),• ; the coefficient ( \ would then have been a positive power series in m, with a numerical factor for each power of m, which factor is always the ratio of two integers. The actual arrangement of e p+p ', e' 1 '', . . . inside ( ), is immaterial, but we shall, in general, retain the arrangement of § 14. Example. — The numerical part of the coefficient of £ 3+2m - 2 « is denoted by 0/y&' 2 a)j. which 1 ias the characteristic e'Va in ut~ l For here. p = o, p'=o : interchanged :2, /=i ; q=2, q'=o with the indices of L I: The series of terms in v.t~ x which will be found along with this term, are those of the same characteristic which are ■I 62 Mr. Ernest W. Brown, Theory of the obtained by putting f 1 for K and whose indices differ from the given index of K by even whole numbers. See § 27 below. They are ae'% 2 aS ii [(^V^ 2 4 l f 2f!+2m_2g +(W 3 ^«) i ,r i - 2m+2g ], where 2 i i =±i, ±3, ±5, . . . 16. Notation for terms with a given Characteristic or of a given Order. — It will frequently be convenient to specify such terms in a brief manner ; this may be done by means of a suffix attached to u, s, z. Thus all terms in u with characteristic e 2 may be denoted by u e >, those with characteristic (fa by u e , a , and generally, those with characteristic ^ by u K , The terms of a given order are denoted by numerical suffixes. Thus. u 3 denotes all terms of the third order ; u , those of zero order ; and so on. Section (v). Method of Solution. Preparation of the Equations. 17. The general feature of the method consists in the successive determination of the terms of orders o, 1, 2, . . . with respect to powers and products of e, ef, k, a in the coordinates u, z. As will be seen from the results of Section (ii), there are two methods of procedure — one by the use of the equations (3), (4), and the other by the use of the homogeneous equations (6), (7), (8). At the same time it may be stated that we need by no means confine ourselves to either of the two sets of equations, but use one or the other or both as may seem most convenient for the particular class of inequalities under consideration at any stage of the approximations. 18. Terms of Order Zero. — These terms, the coefficients of which are functions of m only, constitute a closed orbit with reference to the moving axes which is really the primary or " intermediate " orbit in the same sense as the elliptic orbit of the older theories ; it may be called the Variation Curve since its principal periodic term is known as the " Variation." According to the notation of the last Section it is given by u^=B,%aff\ So^aSflL.iC* (") These values u , s Q of u, s constitute a particular solution of the equation (#4-m)% + -V 2 M + lm 2 s- K v=o (12) X ' 2 2 ft 6 Motion of the Moon. 63 or, in the homogeneous form, of the equations, D\u8) — Du . Bs — 2m(uDs—sDu) + 1m\u + sf= C" D(uDs— sl>u — 2 raws) + 2-m\ii? — s' 2 ) = o (12') since here, Si 1 =o, 2=0. The constant C is a function of a, m, while a is a function of E+M, n, m. The values of u, s being substituted in the equations of either form (by preference, the latter), we obtain a number of equations of condition which suffice to determine aa t . In consequence of the presence of a, one of the a t is arbitrary; we put a =i. The whole theory of these terms and the numerical results have been fully worked out by Dr. Hill in his paper " Researches in the Lunar Theory" (Amer. Jour. Math. vol. i.) ; the results will be given in Chap. II. 19. Terms of the First Order. — We put where u=u + u 1 , z=z u and neglect powers and products u u s t , z u above the first. In the general equations of Section (ii) we pxit 53j=o, s=o, whenw 1 =M e ; Q, 1 =b> 2 , s=o, when u 1 =u l: , ; Q l =(j> s , 3=0, when u 1 =u^ ; g >l =o,u l =o, 'whenz l =z k . Further, when Si x =o). 2 we neglect e' 2 and higher powers, and when Sl x =a> 3 we put </ = o; in both cases u , s may be put for u, s in Q v The right-hand members of the equations thus consist entirely of known terms. Putting u=u li +u l . s=s + s u z=z L , and expanding the last terms of the left-hand members, it is easily seen that the equations (3), (4) for u u z t become D^ l -2Mz l =o ) ' where, putting p 2 = ? 'o'o, (13) 2 2Po J J7=%F,t 2! = 3 -m 2 +• 3 KM o'%;i I 2 2 f , a ' 1 ( l 4) ■■■n 64 Mr. Ernest W. Brown, Theory of the and A consists of a ^-series with known coefficients, and \=e' or a. When Ul =u e , we have A = o. The sets of inequalities corresponding to u e , it e -, u a , z k can evidently be separately determined. In each case the appropriate expres- sions for u h s x , z u are substituted, and equations of condition for the unknown coefficients are obtained by equating the coefficients of the various powers of £ to zero ; they are then solved by continued approximation. In the equation for u e , the questions of the arbitrary constant of eccen- tricity and of the motion of the perigee arise, and in that for z k , the arbitrary constant of latitude and the motion of the node. ; these will be dealt with in their proper place (§§ 25, 26). The homogeneous forms of the equations may be considered in a similar manner. I do not give the developments here as they may be easily inferred from the more general treatment in § 22 of this Section. Further infor- mation is given in chapter xi. of the " Treatise on the Lunar Theory " and in the papers on the Parallactic and Elliptic Inequalities (Amer. Jour. Math, vols, xiv., xv.). The numerical results for the terms of the first order are contained in Chapter III below. 20. Terms of the Second and Higher Orders. Development of the Equation* ( 3 ) ? ( 4 )_ The terms of the first order having been obtained, we proceed to show generally how the terms of any order and characteristic may be found when those of lower orders have been calculated. We shall deal with both sets of equations as either may be useful in certain cases. In this section the equations (3), (4) are considered. They may be written, ™ + m)«u+-Vur 4 3 m*»£ . f 1 - -""f-^-S" 1 -^ • • (15) v 2 2 f l OS m^ m ^- K "-- ld ^-i (16) 2 OZ Suppose that it be required to determine all terms having the cha- racteristic X, say u M in u, and those of characteristic* X, say z k , in z. * Of course X can never be the same in the s-equation as it is in the w-equatioii. Conse- quently, as terms of different characteristics are never found in the same set of equations of condition, the equations (15), (16), are never considered together in finding any particular set of coefficients. Motion of the Moon. 6 c We put u=u (l + S l i^ + u„ 2=%, in (15), u=u + S,u^ z=%z^ + z x , in (16). Here 2u„ S\ contain all terms in u, z, except u , u kt z K , which will contribute to give terms with characteristic \. These expressions are substituted in the equations, which are then expanded according to powers of u-ti 0) z, all powers of u K , z k above the first and products of n k , z k with u M z^ being, of course, omitted. It will be remembered that r 2 = m + z\ Choosing out the terms which may produce terms with the characteristic X we find =Part, char". X, in \ -t l {D* + 2mD)3 < u -^if 1 L os Po 3 (8V. u J S\ s J 4 u e Q 2\ Po J -5 ^y_3s^sA :! _ 9 pOr%_ IS /s^y ^ i6Vm / i6V*oi i6\w / s i6\s ) u 4\ Po J «o 4 \ Po / «o + (] - -. - (17) =Part, ehar c . A, in \ -tr\D°-+2mD)%u -^pJ £-1 4 4 J + ' ' J - - <i7') Z> 2 s J -2i¥s 1 =Part, char". A, in l-D^z- 1 8 J^ L * 2 cz + _!Lr_3 2 ?M ^«; + ?M Po' L 2 p v w « y + %« i5^y + i 5 /5«,y 9 5^ %l_3/s^y Po I 8 V «o / 8 ^ s () y 4 Mi) Sq J 2 V po / Royal Astron. Soc, Vol. LITE. ]] ... ... (I S) 66 Mr. Ekxest W. Brown, Theory of the =Part, char . \, in -D^s, 108, 2 03 ... (i where P=%P£ %i B=%Bff' a KM °^ ' Q = 2,Q^=a Po' 3f-3 T=^ l TX ii ^^ KU °"' Ci (><> P=XP_ff ! , Po" R=%fi^' (i9) etc. the values of R, R being obtained from those of P, R by interchanging u,i \ 6'oC, that is, by putting i/£ for £.* 21. Some remarks on these equations are necessary. In the first place, the left-hand members of (3), (4) being linear with respect to u, s, and to 2 respectively (exception being made of the terms containing k ; these will be considered immediately), the parts Uo + Xiip of u and % of z cannot contri- bute anything to terms with characteristic X. as far as the coefficients of these periodic terms are concerned. But the operators B 2 , D cause the coefficients .in u tl , z^ to be respectively multiplied by factors of the forms (2i+pc\-rmiqgf, (2i+pc + rm + qg), and c, g contain powers and products of e 2 , e' 2 , F, a 2 . Hence it will be necessary in some of the terms, whose orders are higher than the second, to include B\, Du„ B 2 z (J , in the equations. In o-eneral, all the unknowns are contained in the left-hand members of (17) or (17') and (18) or (18'), while the terms of the right-hand members are entirely known. Exception to the last statement only holds when we are determining an unknown part of c or g. It must be remembered that, in * In the Investigations, p. 327, where those expressions are used, one or two errors must bo noted. The factor a (there called a ) is omitted from the values of P, P, Q and the factor a* from those of R, R, T, S. The notations for B, R should be interchanged, that is R_ t put for /)';, to bring them into uniformity with those for P, P. Motion of the Moon. 67 reality, c and g are supposed to represent the complete values at the outset, but that, in forming the equations of condition, all terms of higher orders in the values of c, g, than those actually under consideration are neglected. Hence an unknown part of c or g will, in certain cases, arise from the terms containing the operator I) in the right-hand members. These cases are more fully considered in the next Section. As to the terms involving SZ h since Q t is of the first order at least, apart from the order of the terms in u, we can evidently substitute u, + Xu lx for v and %z^ for z therein. The rest of the terms arise from the expansions of ^I*, ^ in powers of %, %• Those containing the first powers of these quantities are omitted, since they evidently cannot produce terms with the characteristic X. It will be noticed that the terms are written in two ways, (17) ( 18) and (17'), (18'). } If we take the first forms, namely (17), (18), for calculation, the pro- cess is to calculate each Xuju a , XzJp Q (which consists in a single easy multiplication of each u^ 1 by . % £/ p * and of each z >L by i/p Q ), to form the various products inside the parentheses, and finally to multiply the resulting series by Kii^/p* and K /p* in the respective equations. For this purpose the values of a Po' ku £ a/V expressed as even-power series in £ are given at the end of Chapter II. If we take the second forms, namely, (if), (18'), the various multipli- cations which have to be made are evident, and the values of P, Q, . . necessary for this, expressed as power series in £, are given in the same place. There can be little doubt that the first forms give shorter calculations, and they have a further advantage in the fact that the only trouble necessary for transferring to polar coordinates consists in the calculations of powers of XuJuq, Xzjpo (see Section ix), and this labour will therefore have already been finished. Indeed, the second forms would not have been given here at all, were it not for the fact that I failed to see the great advantage of the first forms for the higher inequalities, and consequently used the second forms in 68 Mr. Ernest W. Brown, Theory of the the calculation of all the second order inequalities and of a few of the third order.* A great advantage of these equations is that the chief labour— the multi- plication of series— can be easily arranged for an ordinary computer, and much of the merely mechanical labour may thereby be distributed (see Sect. viii). The numerical results for terms of the second order are given in Chapter IV. below ; those for terms of the third and higher orders will be given in chapters to be published hereafter. 22. First and Higher Order Terms. Homogeneous Equations.— The separation of the homogeneous equations into known and unknown parts is effected in a similar manner. The substitutions for u, z are the same as in § 20. If the homogeneous equations be used for actually finding the co- efficients, the forms obtained in equations (20), (21), (22) are of no assistance, except as a guide ; this will be evident by a glance at §§ 32-36. They have been, however, almost exclusively used for verifying the results obtained from the equations of § 20, and they are given mainly for that purpose here. They are 2)2 ( M(A + 8oU j _ Bu Ds x - Ds Du x - 2 m(u Ds x - s Du x + u x Ds - s x Du ) + lm 2 (u + s )(u x + s x ) 2 =Part, char=. A, in ["<?'- S( ? + iH + ^S 1) L 3= 2 -(D%) 2 - 3 m 2 (%) 2 }l ... ... ... (20) 2 J j D(u Ds x -s Du x + u x Ds - s x Du - 2mu s x — 2ms Q u x ) + T,m%u Q u x —s s x ) =Part, char*. X, in [&&±-u d ®-± ' |_ as ou (*0 U V, * From the remarks just made, it might have seemed an improvement to put u s=s v, z= Po w and to find v, v, w only. This, however, only throws some of the labour of forming u» /w , etc., on to the solution of the equations of condition ; the latter process is far less mechanical than the former and much more liable to error, and there will be no saving of labour. Motion of the Moon. 69 D(u JDz x -s A Du ) - 2mz x Dic - voSi a z x —^mh k (u + s ) 2 2 oz =Part, char=. A, in fs °&-l-J L c« s - {Dtfu, . Dtz^-1%^ . D% Ul )— 2m% . DSt^-m 2 ^. % (22) As before, the left-hand members contain the unknown terms. The right-hand members consist entirely of known terms, except when new parts of the motions of perigee or node are under consideration. In certain cases it will be necessary to suppose u , s to be included in %u M Xs^ ; see the remarks at the beginning of § 21. In the terms involving £1 ; we may substitute u^ + Xu^ Zz^ for u, z, respectively ; the method by which the calculations are to be actually performed will be given in Section (vii). In the case of the first order terms, Zu^, Xz^ are both zero, and the limitations of a t are the same as those given in § 19. Section (vi). Definitions of the Arbitrary Constants. 23. Of the six arbitrary constants of the solution, three have already been defined, namely, the three angular constants in the arguments D, I, F. A fourth, n, has also been defined as the observed mean motion. It remains to give an exact definition to a, the linear constant (which replaces the mass E+M), and to e, k, the constants of eccentricity and inclination. 24. Definition of a. — In elliptic motion, a is defined by means of the relation n 2 a 3 =E+M. For many purposes this is much the most convenient definition even when we proceed to determine the solar inequalities in the lunar theory. But in the theory developed here, the calculations may be materially shortened by a different definition of a. The value of u is given by i( £ l =aSjffl(£ 2i j 1 = 0, From the form of this it is evident that either a or a may be chosen to be anything we wish. The most convenient definition is obtained by putting jo Mr. Ernest W. Brown, Theory of the so that ,E+3fy /{m) (23) where/ (m) is a function of m which, in the case of the Moon, is very nearly unity. The definition must now be extended so as to cover the case when inequalities of any order are being considered. The general form of all the inequalities which have arguments of the form £ 2 * (21 even), are given by uf l =&%la i + 2(#'£' r 'V^"''^^" / « 2 '') i e 2p 'e' 2r 'k 2 «'a'- s ']C 2i where i—o, ±l, ±2, , . . ; p', q', r', s'=o, i, 2, . . , (except p'=q'=r' = s'=o) ; and £ ( )i denotes the sum of all such terms for the values of p', q f , r', s', given. The coefficient of £° in u%~\ by means of which a is to be defined, is therefore, since a =i, a[i + %(^'e v v r 'ri' r 'k'''k v a 2 '') e v e' v k v a 2s '], =a(i + >), suppose, so that i' is a small quantity of the second order at least. There are two practical methods of defining a, each of which has its use according to the equations we employ. One of these is to so define a that every term in v is zero, and therefore that the coefficient of £° is represented by a at every stage of the approxima- tions. This definition requires the determination of some additional terms to a whenever we are finding inequalities of the form Z? 1 . If we are using the homogeneous equations, this is undoubtedly the best definition, for then the determination of the additional terms in a can be left till the end of the work, and as a only appears in the parallax and not in the longitude and latitude, a very great degree of accuracy in its value is not required. In using the equations (3), (4), however, it would cause inconvenience as we should then have to find a further approximation to a at each step. As the latter equa- tions are those mainly used here, we shall adopt another definition better adapted to the calculations. This second definition (which we shall use below) is to give to a the meaning which it receives from the intermediate orbit only and to retain it throughout. Thus a is defined by the equation (23) making &{(E+ ]\[)/n 2 }~ h Motion of the Moon. 71 a numerical constant which never alters. The coefficients v are now no longer zero, but are definite functions of e 2 , e n , k 2 , a 2 , being determined along with the other coefficients of £ 2 ' in the ordinary way. This definition is necessary because of the want of homogeneity of the equations (3), (4).* In finding the parallax from the value of u, all that will be necessary will be to find a/r and then to multiply all the terms by a numerical quantity (which approaches unity very closely) in order to obtain [{E+ M)/n 2 f/r— the quantity usually obtained by lunar theorists. Hence, the linear constant a is defined to be the coefficient of £° in w £ -1 , where m £ -1 represents the intermediate orbit or variation curve only. Its value is given in Chap. II., and it retains this value throughout the whole of the approxi- mations. 25. Definition of e. The first of equations (13) for the determination of the inequalities depending on the first power of e is, since A=o, £-\D + m)X + Mu.t 1 + Ns£= o, the solution of which is obtained by assuming M6 r i =ae2 i (^ +0 + <r 3i " c ), i=o, ±1, ±2, . . . When the substitution is made, and the coefficients of the various powers of l 2i±a equated to zero, we obtain a series of equations of condition for the determination of the unknowns e i5 e'*, c, which, are homogeneous and of the first degree with respect to e„ e' t . The determination of c is made by con- sidering the necessary relation which must exist between these equations ; it is actually found by a different method (see § 28 (b)), and we suppose it known. One of the ee t , ee'*, is arbitrary. The values of e t , e' t may all be made definite by taking e as the arbitrary constant and putting e n — e'n = I. * This is the definition intended in the remarks on p. 343 of the Investigations. A want of clearness in the statement of the definition in that paper has caused a misapprehension of its meaning. Mr. P. H. Cowell in his paper " On the Inelinational Terms " (Amer. Jour. Math. vol. xviii.) uses the homogeneous equations for the determination of the coefficients (kh') b and naturally finds it more convenient to put (k/c') =o. He, however, does not find the addition to the value of a (there called a ) ; this requires a reference to one of the equations containing k. The definition is, I hope, made quite clear in § 24 above. 72 Mr. Ernest W. Brown, Theory of the The coefficients of £°, £-° in u(r l are aee , aee ', respectively. Since, by equations (9), u£- 1 =p cos (V— nt— t) + ip sin (V—nt — t), the coefficient of a sin I in p sin ( V-nt-e) will be * e( e o- £ o')= e ' by the use of the assumed relation. The value of e thus defined is very nearly twice the constant of eccentricity used by Delaunay. The general form of all the terms in uC 1 , which involve £ c , £~ c , is (§ 14), + (ei'V^yV ''^''' a2< \U ' c ] e v e' 2 '"k V]. The definition of e must be extended so as to cover all these terms. It has been found most convenient to define it to be such that the coefficients of £ c and £~ e in the above expression are equal, except when p'=f = r'=s' = o, when we have already defined it by making their difference unity. Denot- ing, for a moment, each of these equal coefficients by /3, so that the terms containing £ c , £ _c in u^ are given by it is evident that the coefficient of a sin I in p sin ( V—nt-t) will be e(e — e ') =e. Hence the constant of eccentricity e is defined to be the coefficient of a sin I in the find!, expression of p sin (V— nt—e) as a sum of periodic terms, where V—nt—e is the difference of the true and mean longitudes and p is the projection of the Moon's radius vector on the plane of reference. 26. Definition of k. The second of equations (13) for the determination of £ k is B%- 2 Mz u =o, in which we substitute where k' i — — k_ i . * This definition, for the terms with characteristic e, is the same as that which I adopted in " The Elliptic Inequalities " (Amer. Jour. Math. vol. xv. p. 261) ; e is there called Y t} . Motion of the Moon. y. The constant k is now defined to be such, that k = — A' =i, so that 2k is the coefficient of a sin F in the expression of z as a sum of periodic terms. The constant k differs little from Delaunay's constant y. The general form of all inequalities containing only £ ±g in zi is ak2„ . s ,( £ i>y "V y >'k l + "'k'"'a h \e^e ' 2r k 2 'V'(£« - £"«), and the definition of k must be extended so as to cover all these terms. AVe now define it to be such that (€*V*yy r 'k l + «l'«'a 2s ') = o, for all values of p', </, r, s\ except for p'=tf =)'={/ =o, when we have already put () equal to unity. It is to be remembered that if we interchan o-e the accents and the sign of i, the coefficient merely changes sign ; hence the co- efficient corresponding to that just written down is also zero. Hence, the constant of latitude k is defined to he the coefficient of 2a sin F in the expression of z as a sum of periodic terms. Section (yii).— Solution of the Equations of Conditio?!. Motions of the Perigee and Node. It will here be necessary to divide up the subject according as we are treating the equation (17) for u, the equation (18) for z, the homogeneous equations (20), (21) for u, or the homogeneous equation (22) for z. 27. The Equation (17) for u. The general type of equation for the terms of characteristic X in u is £-\D + rnfu, + MuJT>- + A\£= a,U (24) where it is to be remembered that 31, N are known even-power series in £ with numerical coefficients (equations (14)) and A contains the known terms with characteristic X arising from the right-hand member. In only one case do the latter terms contain an unknown quantity ; this case, which involves the determination of a part of the motion of the perisree, will be treated in § 28 (Jj). Royal Asthox 80c, Vol. LIII. m 74 Mr. Ernest W. Brown, Theory of the Of the terms with characteristic X, suppose that we require to know those in w£ _1 which involve £ 2,±T , where t is one of the values of ±pc±rm±2g , g, the right-hand member containing such terms. Let, therefore, these terms in A be denoted by A=% i (A i ^ + A'ff^). We substitute M J £- l =aXS i (A..£*"+*'<£*-')- ... (25) where \, \' ( are the unknown coefficients to be found, and equate to zero the coefficients of £ 2 ' +T , £~ 2i '~ T . The result is (2i + r+i+m)% + ^ j M J X i _ j + 2 J N J X' J . l =A i } (2i + T-i-mY\ , ^ + 2 j M J \'^.. J + 2jA T j\j +i =A'„ i ) where j=o, ±1, —2, . . . and 21 either =0, ±2, ±4, . . . or =±1, ±3, ±5, . . . Since r multiplies X; and \'_ t it is evident that the complete values of c, g on the left may be replaced by their values c , g u which are functions of m only. The 3f ( , N f have quickly decreasing values for increasing values of i (see Chapter II.) and, in general, the same remark applies to A ( , A'_ t , \„ X/_ t -. The equations may thus be solved by continued approximation. The unknown terms of lowest order in the equations (26) are [(zi + T+ 1 +mf + M ]X i + F XL i , F \; 4 [(21 + r - 1 - m) 2 + M ]\' .; , respectively. The equations (26) are therefore those of principal importance in finding X f , \'_ f . When we solve them as two simultaneous equations to find \ i} \'_ u the common divisor is [(24 + T+ i+mf+M ] [(2t + T- 1 -m.f + M ]— ¥ " ; and, by the results contained in Part III. of the Investigations, it will be seen that this is very nearly equal to (2i + Tf[(2i + Tf-C Z ] (27) Motion of the Moon. 75 If we had eliminated all the other unknowns from the equations (26), this expression would have been a factor of the divisor, the other factor being- very nearly unity. In considering the solution it is then only necessary to treat the cases where the expression (27) becomes small. 28, There are four special cases to consider — namely, the cases when either factor of (27) is zero or small. We recall (§ 14) that when inequalities involving odd powers of a are under consideration, 21 is an odd positive or negative integer. It is unnecessary to prove many of the state- ments made below ; their truth, if not evident, can easily be demonstrated. (a) The case 2i+r=o. Here we must have t=o, i=o, owing to the incommensurability of c, g, m, 1. It is a question of determining coefficients of £ 2! ' ; there are no coefficients \\ and the two equations coalesce into one which is of principal importance in determining X . (b) The case 2i + t= ±c . Motion of the Perigee. Here i=o, r=c ; t= — c is the same case as t=c as we consider £ c , £~ c together. In this case A con- tains an unknown quantity — namely, the part of the motion of the perigee which has the characteristic X/e, say, c A/e , and it will be found possible to put where B t , b f , B' i} b\ are entirely known. It will be found also that b t , b \ are always the same whatever X may be. Substitute these values for A i} A' _ t in (26). Multiply the first equation by e ( , the second by e'._,-, and take the sum for all values of i. Since t=c , we find 2l( 2 i + c + 1 + m) 2 ,V, + ( 2 i + c - 1 -m) 2 \'_ t e'-J + VfiMAk^ + X'_ w£ '-i) + ZiVW-A + W-<) = 2 i [B i e i + B^^ + c x , e (b i c i + b'_^^)] (28) But for the terms with characteristic e, we have (21 + c + 1 + m)% + VjM fi ^ + XjF/j^=o, (2i + c — i- my_i + 2jMjt'_ M + ZjNfy, ,= o. Substituting these in the previous equation, it is easily seen* that the left- hand member vanishes and therefore the right-hand member of (28) equated to zero determines c A/e . (See Investigations, p. 336). * For we have 'SfiM i X i ^ J € i ='SfiM__Jk i e iH and M H =M p etc. 76 Mr. Ernest W. Brown, Theory of the When the value of this quantity has been found, the equations (26) may be solved by continued approximation, all the \ { , \ f ( being expressed in terms of X , X'f, One of the two equations for \ , X' can then be deduced from the other. An arbitrary relation connects the X ; , \',. We have already (§25) settled this relation to be such that X =X' . The determination of a new part of the motion of the perigee thus goes with a more accurate defini- tion of the constant of eccentricity. The manner in which c may be best obtained is fully discussed by Dr. Hill in his paper " On the Part of the Motion of the Lunar Perigee, &c." (Acta Math. vol. viii.). Its value is there found to fifteen places of decimals. The parts of c which have the characteristics e 2 , e' 2 , k 2 , a 2 properly belong to Chapter V. (" Inequalities of the Third Order ") of this memoir. The calcula- tions, however, have been advanced in this particular direction so that c e , c e », c k * might be obtained ; they will be found in the appendix to Chapter IY. (c) The case 2i + r small compared with unity. The inequalities which are of hnq period compared with the lunar month. A troublesome defect of the method arises here. The divisor (27) contains the square of 21 + t, while the corresponding coefficients \, X'_, are in general of the same order of magnitude as A t /(2i+r). The reason of this is easily seen on solving the equations : one of them, in fact, generally differs from being deducible from the other by a quantity which is of the order of magnitude ii + t. This is illustrated in a striking manner by the long-period inequality whose argument is 2F-2Z (which is one of the most troublesome in any method). Here 2*'+T=2g -2c = +-0272, nearly, and therefore (2z' + t) 2 = + -00074, while the corresponding numerical coefficients are of the order of magnitude unity. The difficulty in all these cases is best avoided by computing the homogeneous equation (21) for the particular value of 2i+r, and combining it with one of the equations (26). In (2 1 ) the terms divided by (21 + r) 2 have the factor m 2 , and therefore with the same degree of accuracy in the known parts of the equation we are able to obtain X„ \'_,more accurately than if we simply used the two equations (26). (d) The case 21 + t±c„ small compared with unity. The numerous short- period inequalities the wean motions of whose arguments approximate to that of the principal elliptic term ■ e.g. the Evection and the Parallactic Inequality. The method is not in defect here, since the divisor is of the same numerical Motion of the Moon. 77 magnitude as that arising in any other method. It will be noticed that the divisors are smaller according as the periods approach more nearly to that of the principal elliptic term and not to the lunar mean sidereal or synodic periods. 29. When the pair of equations for X;, k'_ t possess a small divisor, the approximations proceed slowly. In some of these cases it is advisable to save labour by finding X i±1 , \' t±1 , in terms of X,-, X'_ t and the known quantities, from the equations with suffixes ±i±i, and to substitute the results in the equations with suffix i before solving the latter. In all cases where the difficulty occurred, it has been avoided by this device. 30. The Equation (18) for z. The course of the argument is very similar to that in §§ 27, 28, and therefore the results will be given briefly. The general type of equation for the terms with characteristic X in z is, by the equation (18), D\-2Mz^&XA l . ... (29) where A represents the known terms. For the terms which involve £ 2,±T , we find A=z t A l (£«--F"-') (30) and substitute ^aXSAKi^'-r*-') ■ (31) since in z we always have X'_,- =— X,. The equations of condition are (2i + rfX i -2^,M j \ i _ i =A i (32) where j=o, ±1, ±2, . . . and 2/ either =0, ±2, ±4, . . . or=±i, ±3, ±5. . . . The equations (32) are solved by continued approximation — that written down being of principal importance in finding X,-. The coefficient of X, is (2i + Tf— 2i)I which, if we had eliminated all the other unknowns, would have been (2i + T)*-g * - (33) multiplied by a numerical factor which is nearly unity. 31. There are only two cases to consider, namely, those in which 21 + t is zero or small compared with unity. Hence no long-period inequalities can 78 Mr. Ernest W. Brown, Theory of the give rise directly to large coefficients in z ; thus the cases (a), (c) of § 28 do not arise ; the cases corresponding to (b), (d) are those numbered (//), (d') below. (//) The case 2i + r=±g . Motion of the Node. Here i=o, T=±g , and A contains an unknown part of the motion of the node of the form g A;k . We find A^JJt + g^bf, where B h b t are entirely known. Substituting for A ( m (32), multiplying the equation by ^ and summing for all values of i, we obtain H 2i + go) 2 * A - 2 ^jMjX^jJc^ Zjlfa + g^Xfifa. But for the terms with characteristic k we have (2i + g Q )%—22,M J k i _ J = o. On substituting this in the previous equation, the left-hand member vanishes and we find g vk =-(2,iB,£<)-s-<SM) (34) When this has been calculated, the equations may be solved by continued approximation. The existence of the relation (34) implies the arbitrariness of one of the \ : this has been defined to be such that \=o (§ 26). The determination of g has been made by Adams (M. N. vol. xxxviii., Coll. Works, vol. i.) and later by Mr. P. H. Cowell {Amer. Jour. Math. vol. xvlii.) where full explanations of the method used will be found. The advance numerical results for g e , g e «, g k =, are given in the Appendix to Chap. IV. below, g k * having been found by Mr. Cowell in the paper just referred to.* * Mr. Cowell objects to the above (which was given in a slightly different form in the Investigations) as a practical method, en account of the supposed length of the calculations. He, however, uses equation (8) to find g k » and the accompanying coefficients (P&')„ while the above method contemplates the use of the equation (4). In the latter, the coefficients A- or B t , b t are determined just as in any other set of inequalities ; the only labour that remains in order to find g x/k is the few minutes' work necessary to calculate the equation (34) above. The answer is, therefore, that the homogeneous equation (8), or the equation (4) should be used to calculate both the coefficients \ and g x;k , not one for the coefficients and the other for g Ak . Motion of the Moon. 79 (d') The case 2i + r±g small compared with unity. The inequalities whose periods are nearly equal to that of the principal term in latitude. The remarks made in § 28 (d) apply also here with one or two evident changes ; they need not, therefore, be repeated. 32. The Homogeneous Equations (6), (7) for u. — Suppose that it be required to determine the terms in ul" 1 with characteristic X and arguments 2i±r, where t is one of the values of ±pc±rm±2qg, those with lower characteristics having been found. It will be necessary here to slightly alter the notation of the last paragraphs by specifying the whole of the argument as well as the characteristic in the notation. Let the particular terms in u-X" 1 which have the arguments 2i±r be denoted by aXS^^' + X., ,(?-) ... (35) The coefficients \ Tii , X_ Tii are the unknowns, and are the same quantities as those previously denoted by X,-, X' ; . The equations being of the second degree with respect to u, s, z, we must consider how terms with characteristic X and arguments 2i±r arise in such expressions as D\us), u 2 , &c. The required terms will be made up of terms with characteristic ju,, arguments ±(22 + 0-), combined with terms with cha- (21 + t — cr), where racteristic v and arguments jJ.V = X. In conformity with the notation for terms with characteristic X, let these terms be expressed by ul~ l y^U t 'r-.,t^'-'+y.-r,iS u -' + ') ) (36) It is evident that these may be made to include the terms (35) by putting [m = \, ct=t in the first, or v=\, cr=o in the second. To obtain the corresponding terms in s, we put i/£ for £. As / receives negative as well as positive values, these may be written (36') So Mr. Ernest W. Brown, Theory of the Whence, for all terms with characteristic X and arguments 2J + t, 8*=a 1, XS„y i ,/*_., tV_ r+ „, -j-t-J?**, etc. (37) Since j, e were supposed to have the same range of values, it is evident that i may receive the values o, ±i, ±2, . . . while 2/ receives either the values o, ±2, ±4, . . . or the values ±1, ±3, ±5, . . . according as X con- tains even or odd powers of a. It is not necessary to specify summation with regard to /u- or v ; it may be understood in the summation with regard to cr. The above expressions are so adjusted that the mme set of values for cr will be available in all of them. In general, for characteristics of orders which give sensible terms in the lunar theory, the number of values of yx, v, cr is quite small. [For example, if X=eV, T=2C—m, we have the following pairs of values for /a, <t respec- tively : 1, o ; e, c ; e', -m ; e 2 , 2c ; ee', c-m; eV, 2C-m. The corresponding values of v are derived from the relation pv=\.] When /x= 1, cr=o, the co- efficient is i(y ; it is that denoted previously by a t . The equations (6), (7) must now be put into the forms which will give results best adapted for numerical calculation when we substitute the expres- sions (37) in them. The first process is the calculation of the terms involving z, Q 1 in the equations ; they are evidently known terms, and the calculations consisting chiefly of multiplications, the latter do not call for special remark. When these terms have been obtained the equation may be put into the form, after integrating (7), L i (v.s)-DuI)s-2m(uDs-sDu) + 9 -m%u + sy=L (38) uDs-sDu-2mus + 3 m 2 D- l (u 2 — « 2 )=A (39) where L contains the known terms arising from z, & h in (6) and DA the known terms arising from 81 in (7). No arbitrary constant is needed in A, for the coefficient of £° always vanishes in (7). Motion, of the Moon. The forms of the terms which go to make up L, A show immediately that, for terms with characteristic X, and arguments 21 + t, we shall have A=a 2 ASA(r i+r +r 2; '~ r )- The left-hand members of (38), (39) are also of similar form ; hence, it is only necessary to equate the coefficients of £ 2J+T to zero in (38), (39) in order to find those of £ 2i±T in utr 1 . Multiply (39) by 2m + 1 and add to (38). The result may be written ]JP(u8)-D(uZr x ) . D(s£)-(i +2m-^m 2 V (40) + 9 m 2 M 2 + ^m 2 (2m+ OD-'^ + ^mV-J m 2 ( 2 m + i)/)" 1 ^) =i + (2m+i)A an equation which replaces (38). 33. Substituting the results (37) in (40), (39), and equating the co- efficients of £ V+T to zero, we find (2J + T) 2 — (2i + <r)(2J — 2i + T — ar) -i — 2m H — m 2 /x„ ti v a _. , (a „ , % ,2m + i\ V4 2 2j+rJ + ('9 m2 _3 m ^±l > \ "l =jZ: . + ( 2m + I )A ; \4 2 2J+T/ J X,i (2j-4i + r-2o— 2ra-2)fi <rii v,_ Ttt _ J - j. 3 m * f, 2 2J + T I J , =A,- (4i) (42) The unknown quantities which are found by means of these equations are given by the values [x = \, ct=t, and /x= i, o-=o. Since the a t are known numerical quantities, the equations are linear with respect to the unknowns X ±r j and they can be solved by the ordinary methods of continued approxi- mation. The equations written down are those of principal importance in finding X ; X_ T _•. The principal terms involving these two quantities are obtained by putting cr=r, i=j ; whence /j,„ i =X r y and iv-t, i_,,—«o =I > cr=o, i=o ; whence // [ , £ =a =i and i',_ r ,,-_,■= X_,,_/- Royal Astron. Soc, Yol. LIII. k 82 Mr. Ernest W. Brown, Theory of the The corresponding terms in the respective equations are [( 2 i + r) 2 -i- 2 m + Jm 2 }(X r ,, + X_ T ,__,), j 2 ' ' I" (43) -( 2 j + T )(\ rii -X_, H ) - (2 + 21B)(\ ,, + X.. t , _j), j The method of solution of the simultaneous equations is therefore evident. We find the sum of the two unknowns from the first equation, and thence, substituting, their difference from the second equation. The greater part of the labour of calculation consists in the computation of the first term of each equation, owing to the fact that the coefficients of h<r,t v <T-r,i-j are different for different values of i,j. The coefficient in question in equation (41) is best written (t 2 -ctt + ct 2 — 1 -2m + 1 lmA +{4f—4ij j- 41 2 ) + (4J—2i)r + (4i — 2J)o: The first term of this remains the same while cr remains the same and when r— cr is put for cr ; the second term is always integral ; the third and fourth terms require only multiplication by integers. Hence, after the first term has been obtained, no logarithmic multiplications are necessary to find the whole set of coefficients corresponding to the different values of i, j. The last remark applies also to the corresponding coefficient in (42). The rest of the numerical coefficients in both equations do not involve /. 34. We can deduce from (41), (42) forms in which the coefficient of a \ Tj is— 1 and that of a X._ T> __ ; - is o, so that there will be no need to solve two simultaneous equations as the final step in each approximation to a pair of the unknowns. The details of the algebraical steps are sufficiently simple, and I merely give an outline.* The expressions (43) show that if we multiply (41), (42) by 2m + 2 — 2J — r, ( 2J + t) 2 — i — 2 m + - m 2 , respectively; add and divide the resulting equation by 2(2J + t)U2J + t)' 2 — i-2m + -m 2 |, * Further details will be found in Chap. XI. (ii) of my Treatise on the Lunar Theory, where the spirit of this method is applied to a particular ease. nm % Motion of the Moon. 83 the coefficient of A._ T> _, will be zero, and that of \ rJ will be — 1. The result is 44) where 2J + T, 2*4-0"] 21 + a- 2(2/ 4- t) 2 — 2 — 4m 4-m 2 4- (2*4-0- — 2 j — r)(2j' + T — 2 — 2 m) ' 2j4-r 2(2; 4- r) 2 — 2 — 4m 4- m 2 r 2 ?4-- 1= 3 m a (2i4-r) 2 -4( 2 i4-r)-2-2(2i4-r4-4)m-9m a lJ '' J 4 (2J4-r) 2 {2(2/4- T ) 2 -2-4m4-ni 2 } / 2 ,- + r \_ 3 m 2 5( 2 i + '-) i '-8( 2 i4-T)4-2-2( io/4-5r-4) m4-9m 2 W '' 4 (2i + r) 2 {2(2i + r) 2 -2-4in4-m 2 } .» (45) (21114-2 — 2J — t)Lj ,4- j(2 i /4-r) 2 — i — 2m4--m 2 4-(2m4-i)(2m4-2 — 2j — t) ;- A,- //,,;•=- (2j'4-t) {2(2J+-) 5 -2-4m4m 3 } ... (46) The equation for V_ T> _,- is obtained from (44) by changing the signs of t, j. 35. The case (0) of § 28 deserves mention in connection with equation (44). It will be noticed that whenever the square of 21 + t appears as a divisor, namely, in the second and third terms of (44) and in Aj/(2J+r), (since A ; already contains it as a divisor), the terms have the factor m 2 . The use of equation (7) or (42) for the calculation of the coefficients of long- period inequalities has been noticed in § 28(c). Thus (41), (42) and (44) are free from the objection remarked in that paragraph. It may be also noticed that the expressions (43) show that when 2J+t is small, the corresponding loss of accuracy in the results arises in the differ- ence of \ Tjl X_ Tj _; and not in their sum. This, translated to polar coordinates, is equivalent to saying that the loss of accuracy in the coefficients of loner- period inequalities is chiefly felt in the longitude and but little in the parallax. In the case of short-period inequalities with small divisors the loss of accuracy falls on both coordinates. See also § 31. With regard to the relative advantages and disadvantages of the equa- tions (3), (4) and the homogeneous equations, from the point of view of actual calculation, it seems, from- the experience gained in using both forms, that on the whole the advantage lies with the former. This is certainly true for inequalities of the second order and very probably for those of the third order 84 Mr.. Ernest W. Brown, Theory of the also. For inequalities of higher orders, the operations in the former are numerous but simple, and capable of continual verification ; those in the latter are rather less numerous but more complicated, more productive of error, less easily verified at the various stages, and are much less easily arranged for a computer. The equations (3), (4) are now being used for the inequalities of the third order. If the homogeneous equations be used, the expansions (41), (42) are pre- ferable in actual calculation to (44). This remark does not, of course, apply to the inequalities of zero order, computed by Dr. Hill in the Researches. 36. The Homogeneous Equation (8) or (8') for z may be treated in a similar manner, and a formula similar to (41) or (42) obtained for it. The calcula- tions and results are much simpler, first on account of the less complicated form of (8) ; and, secondly, on account of the fact that there is only one equation of principal importance for the determination of each unknown, instead of two equations for each pair of unknowns. As the equation (4) will probably be used for all inequalities, I shall not develope the equation (8). The motions of the perigee and node are found by continued approxima- tion along with the unknown coefficients. It is possible to eliminate all the latter and to get single equations for the parts of the motions of the perigee or node corresponding to those given in §§ 28, 31 above ; but this will be no saving of time, as the equations are somewhat complicated. The formula) necessary will therefore not be given here. Section (viii). Calculation and Verification. 37. It has already been stated that one of the chief objects aimed at in the developments given above was the reduction of the calculations to forms which made them as far as possible merely mechanical. This is the case with the right-hand members of (17), (18) or (17'), (18'). In the case of the former, — - r , — 2 , are calculated, once for all, from the values of u ,s . At each stage of the approximations ujn is obtained from the value of u^ by multiplication of uj,' 1 by i/u^ l =s ii £,/p l ?, the last quantity being found by " special values-" The remainder of the calculations of the right-hand mem- bers are then simple multiplications of £ series. In the case of the equations (17'), (18') (which were used for the results of Chap. IV. and for some of Motion of the Moon. those which will be published in Chap. V.) P, P, Q, . , . were calculated by the method of special values from those of u , s ; the remainder of the process is then as before. 38. The plan adopted for the multiplication of any two ^-series, say consists in taking out the logarithms of the a t and arranging them along a slip of paper in the order . . . a 2 , a 1} a , a_ u a_ 2 , . . . ; the logarithms of the /3; are arranged along another slip in the order . . . /3_ 2 , ft_ u fi , yS x , /3 2 . . . . The two slips being placed over one another, the sums of all the logarithms for a given value of; are taken without moving the slips, and they are written down in a column. The number corresponding to each logarithm is then taken from the tables and the results added for each value of j. Thus to find the coefficient of t, i+a+fi in the product, that is ^ a 2 _ s #, the slips are placed so that a 2 falls under /3 , then a x falls under /3 l5 &c. The arrangement of this part of the sheet is then — Values of 3 i 2 — 2 — 1 o «V-;#i log number Sum, < /=2 The process can be thus arranged for a computer, and the mere copying of figures from one sheet to another is very rarely necessary. The result of each multiplication of series is verified by adding the sums for all values of j. The sum should be equal to (X&D^Zfit). 39. The values of the A t , A'_ { in equations (26), (32) are in general carried to the same number of places of decimals for each value of i. In the solution of the equations of condition, large divisors frequently occur for large values of i and the X ; , X',- are obtained to one or two more places of decimals for large values of i. Thus it is in general possible to find B"u A to the same degree of accuracy as the A„ A' { . Exception only occurs in the cases of some long and short period inequalities ; when it occurs, the corresponding values of 86 Mr. Ernest "W. Brown, Theory of the A t , A'i are taken to one or two more places of decimals or the homogeneous equation (7) is used, as explained in § 28 (c). The object of taking the values of \„ \\ to more decimals for large values of i is to render the equation of verification more searching. For veri- fication I use one of the homogeneous equations with £= 1 ; the calculation of it is never very long, and it appears to furnish a good test. See Investiga- tions, p. 343. 40. In the following chapters I give in general two sets of numerical results : First, the values of the right-hand coefficients A i} A'r, and, secondly, the values of X,-, X' t . They are taken exactly as they stood in my manuscripts. Although many of them will not be more than two units wrong in the last place given, the number of the calculations prevents this being said of all. They are intended to be trustworthy as far as the last figure but one in each case. The sums of the numbers in each column are always given, so that any error of transcription or typography may be detected should it occur. All calculations are made at least twice, separated by an interval of time. The general plan, when several hours a day were available, was to have two or three separate sets of calculations proceeding together. Each of these would be carried to a certain stage and, after the lapse of a day or two, they would be gone over again, the errors, if any, corrected, the results verified (whenever this was possible) and they would then be taken a stage further. In this way an error running through several pages of calculations was avoided. Section (ix). Transformation to Polar Coordinates, 41. We have, by equations (9), Section (iv), p exp. (V — nt — e)t,=ul~ l , p exp.—(V—nt — e)i=s£, p tan <f>=z. Hence 2i(F— nt — e)=log tit,' 1 — log sg, r (us + z 2 )* ^tan" 1 -. Then and Motion of the Moon. 87 Let V be the part of V corresponding to the values n , s of u, s and let 2lV =log U (~ l — log S$(, 2 t (V + SVJ=log (wo + Sw^r'-log (s + 2s„K> 2l2 V,=lo g ( I+ ^)-lo g ( I+ ^) . t *0 2\Mo/ 2\«o/ Also tan y _.*%£ 1 - s o4_S,;(a,— a_j)sin 2iD t M { _1 + S ^ 2i(«i + «_,-) cos 2iD (47) (48) The value of V may be calculated by the method of special values from those of a { . The various terms of the right-hand member of (47) will have been already found in the calculation of the inequalities. Whence by addi- tion we obtain the true longitude V. 42. With the same substitutions, we have 2^V1 " 5 (49) where iW-E+M. The right-hand member of this equation is then expanded. The various products will have been found, as before. All that remains to find a/r is to multiply the result by a a a r a ' p (since r =p„), which is found from Chap. IT. 88 Mr. Ernest W. Bkowx, Theory of the We may also use equation (5) of § 7 to find i/r, the constant term being obtained by (49) or by the method contained in Part ii. of the Investi- gations. 43. Finally -hc-mc-)'- and P Po P 3^ ' 2w„ 2s„ 2«^ SSj." I ~j-~ — \~ — r . w s o w o s o . (So) The expansions and multiplications will have all been performed, and thence </> may be easily found. Hence the whole process of transforming to polar coordinates will be first, for the longitude, the addition of certain known series and the calcula- tion of V ; secondly, for the parallax, the addition of known series and a multiplication of the whole by a/p ; thirdly, for the latitude, the addition of known series. CHAPTER II Tekms of Order Zero. Section (i). Values of a { , a. 44. The coefficients of order zero have been obtained by Dr. Hill * to 15 places of decimals. They are given by the particular solution of equa- tions (12) or (12') of § 18. This solution is expressed by Mo ^" 1 =a2 i a i i' 2i , where a =i. The value of m = ri/(n— n') used is m= + -08084 89338 0831 1 6. * "Researches in the Lunar Theory," Aram: Jour. Math., vol. i. pp. 247-249. The coefficient denoted above by aa, is denoted by a, in Dr. Hill's paper. Motion of the Moon. Values of 89 i. a%. 6 + "ooooo ooooo 00007 5 + "OOOOO ooooo 01107 4 + - ooooo 0000 1 75268 3 + 'ooooo 00300 31632 2 + 'OOOOO 58786 56578 1 + "00151 57074 79563 4-i — 1 — -00869 57469 61540 — 2 4- 'ooooo 01637 90486 -3 4- 'ooooo 00024 60393 -4 4- 'ooooo ooooo 12284 -s + "ooooo ooooo 00064 -6 4- "ooooo ooooo ooooo Sum 4- "99282 60356 45842 The relation between a and a where n 3 a s =E+M. is given by ^=4- -99909 31419 75298. This relation will not be required after the end of this Chapter until we come to the deduction of the lunar parallax and the expression of the co- efficients containing a=a/a / in terms of a /a'. Section (ii). — Values of M, N, P, etc. 45. From the results of the previous section we deduce the series for M, JV, P, Q, S, S, T, a a ku c, k u ot' 1 ' Pq ap 3 ' ap/ See Chap. I., §§ 19, 20, Z7- The series for P, Q, ... are obtained from those for P, Q, . . . by putting i/£ for £; this is the same as putting -?' for i in the suffixes of the coefficients. Royal Astron. Soc, Vol. LIII. o go Mr. Ernest W. Brown, Theory of the Values of i. Mi. Ni. s + "OOOOO ooooo 5 + 'OOOOO 00009 4 + "ooooo 00056 5 + 'OOOOO 00824 6 3 + •00000 06029 7 + 'OOOOO 70129 7 2 + '00006 28883 4 + '00054 79401 6 I + •00630 84231 2 + -03686 55i7i 8 O + •58902 22856 4 + 175707 88032 7 — I + •00630 84231 2 + -01078 63527 2 — 2 + •00006 28883 4 + 'OOOOI 25690 4 -3 + •00000 06029 7 + 'OOOOO 00982 3 -4 + •00000 00056 5 + 'OOOOO 00007 6 -5 + 'OOOOO ooooo 5 Sum + •60176 61259 + 1-80529 83776 9 The series M is given by Dr. Hill in his paper just referred to. The series M, N are both given on p. 328 of the Investigations. p=%p i ^ i , p=%p^ u , g=s&£*.* Values of i. • P». Qi- 5 + -ooooo 00005 + 'OOOOO 00020 4 + -ooooo 00465 + -ooooo 01462 3 + '00000 40164 + 'ooooi 04704 2 + '00032 38766 + '00066 73632 1 + '02280 40093 + -03476 15314 + 1-17156 77322 + 1-17132 34260 — 1 + '01 084 18484 — "001 1 2 12092 — 2 + '000 10 24640 + -ooooo 3 J 9 2 3 -3 + 'OOOOO 09526 + -ooooo 00327 —4 + 'OOOOO 00092 + -ooooo 00007 -5 + 'OOOOO OOOOI Sum + 1-20564 49558 , + 1-20564 49557 The series Q will not be required. Motion of the Moon. 9i Values of i. Hi. *. Ti, s + -00000 00016 + -ooooo 00004 + -ooooo 00047 4 + 'OOOOO 01268 + -ooooo 00384 + -ooooo 03170 3 + 'OOOOO 93162 + -ooooo 34249 + -ooooi 96595 2 + -00061 °55 2 ° + -00028 51417 + -00105 79688 1 + -03299 09904 + -02103 0383! + "04494 46487 + I-I7IS9 65 6 55 + 1-17171 87304 + 1-17123 01076 — 1 + -00906 65690 + -02103 0383 1 — -00289 67092 — 2 + -00008 18396 + -00028 5*417 + -ooooo 06964 -3 + -ooooo o7394 + -ooooo 34249 + -ooooo 00136 -4 + -ooooo 00066 + -ooooo 00384 + -ooooo 00006 -5 + -ooooo 0000 1 + -ooooo 00004 Sum... + 1-21435 67072 + i'2i435 67074 + i'2i435 67077 The series P, Q are given on p. 328 of the Investigations. The co- efficient Ri of that memoir is here called i?_,-. The coefficients <Q 3 , Q_ 3 are corrected here, each of them having been diminished by one unit in the ninth place. 46. Values of the coefficients of £ 2i in the expansions of i. a 1' a Po" 5 + •ooooo ooooo I 4 + •ooooo 00012 2 3 — -ooooo 00156 + ■ooooo 01632 9 2 — -ooooo 35 8l 5 + •00002 29OO7 1 - -00151 57497 + •00358 99818 9 + '99997 36392 + •99999 97077 7 — 1 + -00869 54035 + •00358 99818 9 — 2 + -00007 54483 + •00002 29007 -3 + -ooooo 06521 + •ooooo 01632 9 -4 + -ooooo 00056 + •ooooo 00012 2 -5 + 'OOOOO ooooo I j Sum + 1-00722 58019 + ] •00722 58019 9 92 Mr. Ernest W. Brown, Theory of the Values of the coefficients of I 21 in the expansions of i. apo 3 ' 5 + 'OOOOO OOOOI 3 + -ooooo 00000" 4 4 + 'ooooo 00144 3 + 'ooooo 00048 5 3 + 'ooooo 15058 3 + 'ooooo 05751 9 2 ' + '00015 J 7769 1 + "00006 875 1 1 8 I + -01439 I 49° l 8 + '00841 09322 7 O + 1-17141 74324 9 + 1-17144 79701 8 — I + -00242 99016 5 + '00841 09322 7 — 2 + -ooooi 62575 2 + '00006 875 1 1 8 -3 4- -ooooo 01171 8 + 'ooooo 05751 9 -4 + 'ooooo 00008 8 + -ooooo 00048 5 -s + 'ooooo ooooo 1 + -ooooo ooooo 4 Sum 4-1-18840 84972 1 + 1-18840 84972 4 These four series will not be required until Chap. V., as the calculations 'O'f Chap. IV. were made with P, P, Q; for a few of the inequalities of Chap. V. the series B, R, S, T have been used. CHAPTER III Terms of the First Order. 47. The terras of the first order have been treated in Chap. L, § 19. The results contained in this chapter are classified in the following table : — Section. Charaoteribtic. Arguments. Quantities found here. Coefficients. Motions of Args. (i) (ii) (iii) (iv) e e' a k 2J±C 2i±m ±(2i + g) i?i. V'i Co go where '•*> —3s —5 Motion of the Moon. 93 The formula) furnished by equations (15), (i 6) of Chap. I. will be given in each case, although the results may have been otherwise obtained. Refer- ences will be made to all previously published results. Section (i). — Characteristic e. Value of c . 48. The value of c , which is the part of c depending only on m, has been found by Dr. Hill * to 1 5 places of decimals. It is, c = + ro7i58 32774 16012. The equation satisfied by c and the terms with characteristic e is The'solution is tt a £- 1 =aeS,(e J 4 !U+ « + e',?»- c ). The equations of condition for the unknowns having been obtained by substituting the assumed solution in the differential equation, and equating the coefficients of the various powers of £ to zero, we may solve them with the above value of c, so as to give e ; , e' s - in terms of e , e'„. Let £ »=Vo + A e 'o> e';=b' 4 « + /3y o . Values of i. h- ft- 5 4 + 'OOOOO 00005 — 'OOOOO 00006 3 + 'OOOOO 00843 — 'OOOOO 00708 2 + '0000 1 47376 — 'OOOOO 85373 1 + '00308 02927 — '00092 80067 + 1 — 1 + '01999 88763 + '20567 90112 — 2 + 'OOOOI 15205 + '00007 34691 -3 ■ — 'OOOOO 00193 — 'OOOOO 01734 -4 1 — 'OOOOO OOOOI — 'OOOOO 00012 -5 Sum ... 1 + I'023IO 54925 + '20481 56898 ' Motion of the Perigee, etc.," Acta Math. vol. viii. p. 35. 94 Mr. Ernest W. Brown, Theory of the Values of i. w. ft'. 5 — -ooooo 00002 4 — "OOOOO 00029 — -ooooo 00212 3 — 'OOOOO 04039 — -ooooo 29218 2 — -00005 93876 - -00043 20782 I - -01054 68058 — '°7779 5543° + 1 — I — -00108 65960 — -00019 59999 — 2 -f -ooooo 01043 — -ooooo 08618 -3 + 'OOOOO 00024 — -ooooo oooS5 -4 -5 Sum — -01169 3089s + -9 2I 57 25684 The arbitrary constant e is defined (Chap. I., § 25) to be such that «n~«n =1- Either of the two remaining equations of condition (those of principal importance for finding e„, e' ) then gives e 4-c' = — -49679 18022. From these two equations we find e , e' 0) and thence, from the numbers just given, the values of e h e\. Values of i. e i« 1 5 + •ooooo 0000 1 4 + ■ooooo 00005 + •ooooo 00152 3 + •ooooo 00742 + •ooooo 20851 2 + •0000 1 C0977 + •00030 84234 1 + •00146 95307 + •05556 82459 + •25160 40989 — •74839 59011 — 1 — •14889 75297 — •OOOI2 67065 — 2 - •00005 20854 + •00000 06713 -3 + •ooooo 01250 + ■ooooo 00048 -4 -5 + •ooooo 00009 Sum + ■10413 43128 — •69264 31618 Motion of the Moon. 95 I obtained these results by the use of the homogeneous equations.* A different set of values for b t , V u /3„ I3\ will naturally arise if we use the equation at the beginning of this section. One slight error which occurred in the reduction of b' 4 e + y8' 4 eo to the final value of e' 4 was discovered and corrected. The short-period inequality with a small divisor is the "Evection"; the corresponding coefficients are e_ x , e\. Section (ii). — Characteristic e'. 49. The equation is Z-\D + mfu e , + Mu/C l + Ns/(= - »£-' H~\ os In the right-hand member we put s 1 =&> 2 , z=o, u=u , s=s , and neglect powers of d above the first (Chap. I., § 19). Hence, by Chap. L, Sect, (iii), ds 2*0 +" - fr 2 M 2 (- 1 4 The solution is The equations of condition are formed and then solved by continued approximation. Values of i. ? Vi- Vi'- 5 — - ooooo ooooo °3 + 'OOOOO ooooo 24 4 — 'OOOOO 00004 40 -f- 'OOOOO 00030 59 3 — 'OOOOO 00572 63 + 'OOOOO 03956 99 2 — '00000 76025 41 + '00005 22794 42 r — •00103 48418 2 +•00695 08210 5 — •09186 93227 + •09869 89451 — 1 —•03636 42746 8 4- "00448 82585 5 — 2 + 'OOOOO 17438 21 — "OOOOO 01475 00 -3 + - O0000 00322 23 — 'OOOOO 00041 9 6 -4 + 'ooooo 00002 08 — 'OOOOO ooooo 29 -5 + 'OOOOO ooooo 01 Sum — •12927 43 2 3 2 + •11019 °55 12 * " The Elliptic Inequalities in the Lunar Theory," Amer. Jour. Math. vol. xv. pp. 259-261. 9 6 Mr. Ernest W. Brown, Theory of tin The above method was used to calculate all these coefficients. The long period inequality with a small divisor is the " Annual Equation," having the coefficients i^, i/ - The method of § 29 was used in the approximations to these two coefficients. The values of the corresponding terms in the true longitude have been given in a note in the Monthly Notices, vol. liv. p. 471. Section (iii). Characteristic a=a/V. 50. The equation is t l (D + m) V + Mu£-* + Ns£= - -j^-'- 1 ] ■ x ' as In the right-hand member we put £3 1 =<y 3 , z=o, </=o, u=u , s=s . Hence, by Chap. I., Sect, (iii), as 4 a 2 |_2 2 J The solution is where 2 i=±i, ±3, ±5 Values of 22. («)«• 9 7 S 3 1 — 1 -3 1 ~s I ~ 7 I ~ 9 + •00000 OOOOI + •00000 00072 + - ooooo 04839 — -00005 88448 -•06417 03547 + •17899 19628 — '00293 82096 — •00000 18325 j — 'ooooo 00029 ! • Sum + •11182 32095 Motion of the Moon. 97 These coefficients I found to seven places of decimals in a paper " On the Parallactic Inequalities in the Lunar Theory " * by the use of the homogeneous equations. They have been recalculated and extended to ten places by the above method ; errors of one unit only in the sixth places of decimals in the values of a_ h a, were detected. Dr. Hill in his paper " On the Periodic Solution, &c.,"f using my former values as a first approximation has also recalculated these terms to a high degree of accuracy by a totally different method. The short-period inequality with a small divisor is the " Parallactic Inequality," having the coefficients a v a„j. Section (iv). Characteristic k. Value of g . 51. The part of the value of g which depends on m only, namely g , has been obtained by Professor J. C. Adams and Mr. P. H. Cowell (see the references in § 31). The latter finds g = 1-085 17 14265 58. The slightly different result obtained by Adams is due to the use of a different value for m. The equation giving g and the terms with characteristic k is DH k — 2Mz k =o. The solution is The constant k is defined (Chap. I., § 26) to be such that k =i. * Amer. Jour. Math. vol. xiv. p. 157. A different notation is there used. f Astron. Jour. vol. xv. pp. 137-143. Dr. Hill informs me that the large correction which he obtained to my value of the coefficient of the Parallactic Inequality n longitude, amounting to 5 units in the fifth place of decimals in the value of a h — a_ s , was due to a slight error in reducing them to his form. As the resulting value was only used as a first approxi- mation, his final results are, of course, correct. Royal Astron. Soc, Vol. LIII. p 98 Mr. Ernest W. Brown, Theory of the Mr. Cowell finds the following values (be. cit. p. 113) o Values of i. *,. 5 4- -ooooo ooooo 01 4 + -ooooo 0000 1 75 3 + 'OOOOO 00299 82 2 + 'OOOOO 58673 61 1 + -00151 22192 28 + 1 — 1 — -03698 393*3 94 ■ — 2 — -00004 65750 01 1 -3 — -ooooo 01755 37 i -4 — -ooooo 00008 87 -5 — -ooooo ooooo °5 Sum + -96448 74339 23 I have verified these results by means of the homogeneous equation (7) with a 1=0, by putting £= + 1, — 1, successively, after the substitution of the values in the equation. The short-period term with a small divisor is that having the numerical coefficient k_ x . CHAPTER IV Teems oe the Second Ordek. Section (i). Formula. 52. The general type of the equation for u for terms with characteristic X, and arguments 21 ±r, is, by Chap. I., § 20, l-\D-{-jnfu > .+ Mu x t,- 1 + NsJ:=^\A ... . . ... (1) where and A=ti (A ( F^+A't C M - r ) when.r^o, yl=Sj A i l 2i when t=o. Motion of the Moon. Here, by Chap. I., equation (i7')> aX4=Part, char c . X, in 99 OS +| p (s», n* + - 8 S - e (% o 2 + | ^ (s«j (so-3 * (%)' 2 ] - (2) 5« — s k . In all cases s is derived from u by putting i/£ for £. The first term of (17') contributes nothing to the terms of the second order. Also, by Chap. L, Sect, (iii), substituting for u, s, z and neglecting powers and products of u^ s^, z^ we have i (* + SsJ a 2 + _(« + 2« ,)&*] a ^ (s 2 +2 So S*„) «3 +| (m 2 + 2 M 2m J C 3 +^ (t6 S + «0 2s„ + «o 2«> 3 j +3 T3S 3 J. IS 16 a' 2 |_i6 16 -* 5 - "o 2 *0 (3) where n 2 , 6 2 take their values as far as e' 2 -when multiplied by w 0) s ; a 2 , 6 2 take their values as far as e' when multiplied by u M s^ ; a 3 , C 3) C 3 take their values as far as e' when multiplied by u , s ; a 3 , c 3 , C 3 are unity when multiplied by u^ s^. The solution is of the form ^(T^aXSAP'^+A',?'-'), when r^o, and u£- l =aX&M*, when t=o (4) (4') The process consists in first finding the series for A in each case, and then, after the substitution of the solution in the differential equation, to form the equations of condition for the unknown coefficients. These are solved by continued approximation, there being one pair of equations of principal importance in finding each pair of coefficients \, \'_ t . The known values c , go of c, g, and the definitions of e, k given in Chap. III. are suffi- cient. The further definition of the linear constant (§ 24) occurs in section? (ii), (iv), (v), (viii) below. ioo Mr. Ernest W. Bkown, Theory of the 53. The general type of the equation for z, for terms with characteristic X and arguments 2% + t, is, by Chap. I., § 20, D % z x — 2Mz x =aXAi (5) where jl=Vi(i* +T -£" M -*)- Here, by Chap. L, equation (18'), aXJ.t=Part, char . A, in —^ \_ 2 OZ '(PSu^ + PSsJ^ (6) 2 a Sw^ssMe + Mj. + M., '%Z l ==Z k . The first term of equation (18') contributes nothing to the terms of the second order. Also, by Chap. I., Sect, (iii), substituting for u, s, z the value? just given and neglecting powers and products of u^, s^, z M 2 oz 2 al - l ^=^A + ^^H + s ) (7) -where b s takes its value as far as (/. The solution is s^aAS^p+'-C-*-') (8) The process is the same as before. There is only one equation of principal importance in finding any coefficient \ t . The known values c , g of c, g and the definitions of the arbitrary constants given in Chap. III. are sufficient. 54. The following table gives the various classes of terms of the second order, with the sections in which they are considered below. Motion of the Moon, 101 where Section. K. Arguments. Type of Coefficients. (H) 6 3 2i±2C, 2i (' 2 ). (*'% («') (iii) ee' 2i±(c+m), 2i±(c — m) («»)> («V). («?'). (*'v) (iv) e' 2 2i±2IIl, 2% (V% (i% (rrf) (v) k 2 2i±2g, 2t (*»), (A' 2 ), (*#) (vi) ea 2i!±C (ea), (c'a) (vii) e'a 2%±m (r?a), (rj'a) (viii) 2 a 21 (a 2 ) (ix) ke ±(2i + g + C ), ±(2»+g-c) ±(Ac), ±(/5*') w ke' ±(2i+g + m), +(2i' + g-m) ±(%), ±(fo)') (xi) ka ±(2%+g) ±.(ka) -.2, 3:4 «i=±i, ±3, ±5 The coefficients in the last column have the suffixes i or i h when the corresponding arguments have them, that is, according as the coefficients do not or do contain the first power of a. It will be remembered that when the characteristic contains an odd power of k (that is, for the terms in z), (&Y)j=-(£e)_ 8 ., (^e) < =-(^e / )_ i , etc. Hence, Sections (ii)-(viii) contain all second order terms in u, Sections (ix)-(xi) all second order terms in z. The results selected for publication are the numerical values of A i; A' t and those of the unknowns X i} X',.. The degree of accuracy to which the various results have been carried depend, first, on the general numerical magnitude of the characteristic ; secondly, for A t , A\, on the cases where 21±t, 2i±r±c, or 2i± T ±s become small ; thirdly, for \, X' t , on the accuracy required for certain terms of hio-her orders. The approximate numerical magnitudes of the characteristics* are given by •xi, ' = •017, k=-o45, a=-oo26. For further remarks on the numerical results, see Chap. I., Sect. (viii). References to previously published results are given in all cases. 102 Mr. Ernest W. Brown, Theory of the Section (ii). Characteristic e 2 . 55. Here \=e 2 , and 21 + t has the values 2«'±2c forming one set of equations of condition and the value 2% forming another set. The values of A corresponding to the two sets are obtained from equation (2) of this chapter, and they are given in the following table :— Values of A. Coefficients of i. £2i + 20. f«-°- c . T 2i - 5 4- -ooooo 0002 4- -ooooo 0397 4- -ooooo 0059 4 4- 'OOOOO 0146 4- -00002 4815 4- -ooooo 4028 3 4- -ooooi 0222 4- -00125 4227 4- -00025 3358 2 4- '00064 6617 + •03959 7603 + •01291 0151 1 + -03321 1631 - -08783 1650 + •41545 8343 4-1-08874 2 S37 4--2ig26 5334 -+6585 5712 — 1 — 'i°743 6306 + -00383 3333 4- -08798 3630 — 2 4- -00888 °S39 4. -00005 3547 4- -00151 7439 -3 + -00015 0316 4- -ooooo 0664 4- -OO0O2 1196 -4 4- -ooooo 2104 4- -ooooo 0008 -(- -ooooo 0263 -s 4- -ooooo 0026 j 4- -ooooo 0003 Sum 4-1-02420 7834 4--i76i9 8278 1 4--05229 2758 The solution is expressed by (Chap. I., Sect, (iv)), vS-^arfs, W +a '+ W"*+ ("V Solving the two sets of equations of condition— namely, that giving ( e 2\ (Y2), j^nd that giving (ee')* — we obtain the values of these coefficients. Motion of the Moon. Values of 103 (« 2 >- <« ,2 )*. (««% 5 + 'OOOOO 00000 1 + 'OOOOO 00049 + 'OOOOO 00004 ° 4 + "OOOOO 0001 I + 'OOOOO 04893 + 'OOOOO 00459 5 » j 3 + 'OOOOO 01135 4 + '00004 84244 + 'OOOOO 47226 4 2 + - OOO0 1 16070 9 + '00428 5788 + '00046 03442 1 + •00112 37013 + •01564 7028 +•03917 99373 + '09402 3537 + •03180 1697 — -I 33" 2689 — 1 —•06517 3271 + '00006 45654 + '01492 2756 — 2 + ■00133 0056 + 'OOOOO 06650 + '00002 21364 -3 + 'OOOOO T7404 + 'OOOOO 00057 4 + 'OOOOO 02603 -4 + 'OOOOO 00260 + 'OOOOO 00000 6 + 'OOOOO 00022 8 -5 + 'OOOOO 00003 + "OOOOO 00000 2 Sum + •03131 75 12 + •05184 8668 -■07852 2483 These coefficients were given to eight places of decimals on pp. 325, 323 of my paper referred to in § 48, having been obtained by the use of the homo- geneous equations. The notation is different. The symbols (e*), (e' 2 ) are there denoted by f/Y 3 ,f/Y 3 respectively ; the symbol (ee'),- used here is not the exact equivalent of 8a,/ F 2 in that paper, owing to the meaning there assigned to a being different from that of a. To compare them we must put («');=(«„ Stti+aj Sa )+1 7 2 a 2 , the terms in the right-hand member being the quantities contained in the paper referred to. An error of one unit in the sixth place of the value of (e 2 )_ 2) or/_ 2 in the paper, was discovered, inducing smaller errors in the other coefficients, All the coefficients have been re-calculated by the method of this memoir and the results, as seen above, extended to nine places of decimals. The long-period inequality with a small divisor is that having the co- efficients (e' 2 )_i, (e /2 )i- This was separately calculated by the homogeneous equations as in the " Elliptic Inequalities " and the results were verified by the equations of condition which the above method furnishes. 104 ^ r - Ernest W. Brown, Theory of the Section (iii). Characteristic ec / . 56. Here 2^+ T has the two sets of values 2^±(c + m) and 2Z±(c-m). The corresponding values of A are given in the following tables. ; ; : Value of A. Coefficients of S 4 3 2 1 o — 1 — 2 -3 Sum i. ("2i+c-!-m i £2i-C- m , i 5 — '00000 0002 + '00000 0204 4 — 'OOOOO 0159 + ■00001 2849 3 —•0000 1 0382 + -00066 4341 2 — •00056 4777 + ■02214 "77 1 —•02154 4824 + ■02587 20248 — •26294 86805 — -08121 00493 — 1 + "04040 239 8 3 — •00205 8170 — 2 + •00196 1854 — -00005 0845 -3 + '00006 349 2 — 'OOOOO °753 -4 + -00000 0984 — 'OOOOO 0009 -5 + 'OOOOO 0013 Sum — •24264 0083 — •03462 9230 Value of A. Coefficients of /2t+C-D + •00000 + 'OOOOO + -00006 + •00341 + •11582 + •23207 — ■02089 —•00037 — 'OOOOO — •00000 — "OOOOO 0016 1031 557° 6192 9253 94821 28787 5417 9929 0149 0002 ^'" c — 'OOOOO — 'OOOOO — 'OOOII — •00414 —•04792 + •09218 + •01113 + -00032 + -ooooo + -ooooo + -ooooo 0032 2037 0038 1602 22751 67678 8493 6072 499S 0066 + ■33011 3168 + ■05148 041 1 Motion of the Moon. 105 The solution is tt^-^aee'S,^),; a+c+m + (c'»,')«<: 2 '^ m + HU 2i+c ' m + («W~° +,B ]» the first two terms forming oae set of equations of condition, and the other two another set. Values of i. ("») 4 - («V),. 5 4 3 2 1 — 1 — 2 -3 -4 -5 — •00000 00016 — •ooooo 01635 — •00001 60440 -■00143 5419 —■09352 2778 — •37910 7012 -•00035 !5° 2 + •00000 14383 + •00000 00149 + '00000 OOOOI + •00000 00023 + "ooooo 02402 + "00002 40337 + •00223 19829 + •16122 2282 + T45 I 5 115 + •00011 3163 — •ooooo 09258 — -ooooo 00095 — "COOOO OOOOI Sum -•47443 1467 + •30874 192 i. («»'), (*'")r 5 + •00000 OOOOI — 'ooooo 00003 4 + '00000 OOIOI — 'ooooo 00346 3 + '00000 10020 — •ooooo 34589 2 + '00009 30969 -'00032 25311 I + •00721 48506 -•02437 4803 + •12769 0229 -•22224 55 2 — I + •03961 720 —•00035 7897 — 2 + •00004 6236 + •00000 62531 -3 — •ooooo 01813 + 'ooooo 00632 —4 — •ooooo 00020 + "ooooo 00006 -5 Sum + •17466 244 -•24729 793 Royal Asteon. Soc, Vol. L1II. T06 Mr. Ernest W. Brown, Theory of the The short period inequalities having small divisors are those with coefficients (e V ) , (*y)o ; («*)_!, (eV)i 5 («/)* («fy)oJ (o/)_i, (^)i- For the purpose of obtaining these with the required accuracy, the corresponding coefficients in A are carried one place further than the rest. The values of A have been computed by both (17), (17') of § 20. The slow approximations to the values of the coefficients with suffix zero were avoided by the method of § 29.* The values of the coefficients of the corresponding terms in the true longitude have been published in a note in the Monthly Notices, Vol. LV. p. 4. Section (iv). Characteristic e n . 57. Here 21 + T has the two sets of values 2«'±2m and 21. The terms are similar in form to those of Section (ii). Values of A. Coefficients of i. £2t+2m_ C ai-Sm, C M - 5 + •00000 003 — "OOOOO OOI 4 + -ooooo 003 + 'OOOOO 166 — 'OOOOO 052 3 + -ooooo 236 + -00008 791 — -O0OO2 893 2 + •00010 062 + -00308 °35 — '00113 096 1 + '00208 790 + •01194 981 — '01521 048 + •01086 3018 + -00684 6387 -•03584 293 — 1 — •08426 889 — •00006 164 + •02561 810 — 2 + •00013 885 + -ooooo 798 — -00005 617 -3 + 'OOOOO 733 + 'OOOOO 019 — -ooooo 228 -4 + 'OOOOO 013 — 'OOOOO 003 -5 Sum ... — •07106 865 + •02191 268 — ■02665 421 * The results for the coefficients in A are not theoretically accurate in the last places of decimals given. The small divisors occurring in the coefficients mentioned and the other divisors are of such a size that the results for the coefficients («;) a &c, are, however, theoretically accurate to the last place given in each case. w Motion of the Moon. The solution is which gives two sets of equations of condition as in Section (ii). Values of 107 i. M,- (V% (rm%. 5 4 3 2 I — I — 2 -3 -4 -s + 'ooooo 00004 + 'ooooo 0035 + •00000 2205 — •00003 2 77 -•05446 177 -•10598 405 + •00001 0669 + •00000 0238 + •00000 00026 + - ooooo 00003 + •00000 00290 + •00000 2938 + •00027 6 5 86 + •02192 232 + ■07221 455 + •00007 2 ^7 + •00000 0016 + -ooooo 0003 — 'OOOOO OOOOI — 'ooooo 00083 — 'ooooo 08202 — •00007 6201 -•00585 014 -•01024 957 + •02515 958 —•ooooo 1878 — •ooooo 0060 — '00000 00005 Sum — •16046 544 + ■09448 911 + -00898 090 The long-period inequality with a small divisor is that having the coefficients (rf) , (y' 2 )o- To obtain it with sufficient accuracy, the homo- geneous equation (7) or (42) of Chap. I. was calculated for 2z'+T=2tn, and combined with one of the equations of condition of principal importance in finding these coefficients by the above method ; the corresponding terms in A are carried one place further. The slowness of the approximations was avoided as before. Section (v). Characteristic k 2 . 58. Here ii + r has the two sets of values, 2r±2gand 21. The forms are similar to those of Sections (ii), (iv). io8 Mr. Ernest W. Brown, Theory of the Values of A. Coefficients of i. C 2i+2 *. £3t-2g. C 2i . 5 + '00000 0001 4- "OOOOO 0005 + -ooooo 0005 4 + "00000 0098 + -ooooo 0373 + -ooooo 0420 3 + -ooooo 8104 4- 'OOOOI 9128 + -00003 0535 2 + - ooo6i 3392 4- -00019 5705 + -00179 3676 4 I + '03948 0728 - -09576 3403 + -05619 8303 O + 175467 4677 + 1-75605 S74S -S'S 1 ^ 8315 — I - -11368 2769 + -02156 4305 + -09211 3861 — 2 + -00119 0891 + '00022 7392 + '00118 5620 -3 + -ooooi 6227 + 'OOOOO 2257 + -ooooi 2707 -4 4- -ooooo 0178 + -ooooo 0022 + -ooooo 0126 -5 + -ooooo 0062 + -ooooo 0001 Sum + 1 '68230 *529 + 1-68230 1529 —3-36460 3061 The solution is t^- 1 =ak 2 2,((^) i C 2i+2g +(A' 2 )^ 2i - 3g + (M')^ 2i ]. Values of i. <**>«• (# 2 V (»'),• 5 + •ooooo ooooo 8 — 'OOOOO ooooo 1 4 + -ooooo OOOOO 2 + •ooooo 00096 — -ooooo 00014 9 3 + -ooooo 00009 + ■ooooo I33 01 — -ooooo 01983 2 + -ooooo 01113 + •00020 4729 — -00002 78210 1 + -oooo I 40450 + •04329 3868 — -00434 42967 + •00165 67611 + •98752 5842 — 1-00079 9130 — 1 — -09302 7702 + ■00150 88256 + -08149 6924 —2 + •00081 6246 + •ooooo 58653 + -00009 25048 -3 + -ooooo 14413 + •ooooo 00300 + -ooooo 03390 —4 + -ooooo 00059 + •ooooo 00001 8 + -ooooo 00016 6 -5 + -ooooo ooooo 4 + -ooooo OOOOO I \ Sum . . . -•09053 9090 + [■03254 0500 - -92358 1678 Motion of the Moon. 109 These coefficients were given to seven places of decimals by Mr. P. H. Cowell on pp. 119, 117 of his paper referred to in § 3 1 above, being obtained by means of the homogeneous equations. With his values as a first approximation, I recalculated and extended them to nine places by the above method. Small errors in (k*)_ u {l J% \ were found. These two coefficients are those of a long-period term with a small divisor. The former was obtained by means of equation (44) of Chap. I., the latter being then found from one of the two equations of condition furnished by the general method used for the rest of the coefficients. The coefficients {kK) t in Mr. Cowell's paper are such that (k¥) =o ; to compare them with those given here, a transformation like that noted at the end of Sect, (ii) of this chapter must be made. The value of Sa in that formula is the (M 7 ) of the table in this section. Section (vi). Characteristic ea. 59. Here 2i + T=2i 1 ±c where 2% is an odd positive or negative integer. It is not necessary to insert the suffix of i x in the tables. Value of A. Coefficients of 2i. f2i + C p-e. 9 + '00000 002 + '00000 010 7 + 'ooooc 053 — '00000 676 5 — •00003 223 -•00135 769 3 — •00682 665 — 'IOOI0 7433 1 -•51730 75492 +•13499 37257 — 1 + •14791 32045 — -I00I2 41023 -3 — •00226 4428 —•0065s 106 -5 — '00141 822 — '00004 061 -7 — •00000 864 — -ooooo 022 -9 — '00000 004 Sum -■37994 400 -•07319 40s The solution is M^-^aeaS^aXP'^+Ce'a)^'-'], 2% odd. no Mr. Ernest W. Browk, Theory of the Values of 2i. («)«• <«'»),. 9 + '00000 00001 + 'OOOOO 00007 7 + "OOOOO O0I2I — 'OOOOO 0210 5 + "ooooo 0067 — - oooo6 845 1 3 — •00018 3543 -•01423 8397 i -•0467s 5060 -•13023 797 — i + •19695 832 — •01226 5219 -3 +•01753 2892 —•00059 1716 -s — •00014 55" — "OOOOO 1318 -7 — -ooooo 0318 — 'OOOOO 000 10 -9 — 'OOOOO 00008 Sum + •16740 685 -•15740 328 The long-period inequality is that having (ea).^ (^0.)$ as coefficients. The homogeneous equation (42) of Chap. I. with ii+r= — i+c was used with one of the two ordinary equations of condition giving these two co- efficients, the other being, as usual, used as a control. The slow progress of the approximations was avoided as before. Section (vii). Characteristic efa. 60. Here 2i + T=2i 1 ±m. The suffix of i x will be omitted. Value of A. Coefficients of 2i c- i+m - fti— m # 9 + 'OOOOO 003 7 + -00000 012 — 'OOOOO 138 5 + -00004 540 -•OOO33 248 3 + '00405 8824 — •O2654 623 1 + ■06010 42164 -•05865 88010 — 1 — -04094 6441 + •01713 45698 -3 -•06584 676 + ■01489 4425 -5 — •00036 75 2 + -0O006 690 -7 — 'OOOOO 211 + 'OOOOO 036 -9 Sum -•04295 427 -•05344 261 Motion of the Moon. in The solution is u e , ^ 1 =ae'aS,-[(i?a) 1 .^ +m + (Va) i (; 2{ - m 3, ' 2% odd. Values of 2i. (i a ) r (V«),. 9 + "0000O 000 10 + "OOOOO 00016 7 + '00000 0123 + 'OOOOO 0126 5 + "00001 8262 + 'OOOOO 8022 3 + •00286 0184 + -00008 9662 i + •51611 841 — -02661 542 — i + •03082 496 — 1-51100 393 -3 — '01661 267 + -00301 4686 -5 — •0000 1 8246 + -ooooo 443i -7 — "00000 0037 + -ooooo 0015 -9 + 'OOOOO 0000 1 + -ooooo 0000 1 Sum +•53319 099 -i'5345° 241 The shorfc-period inequalities are those having the coefficients (i?a)j, (i/ a ) _ 4 ; (^a) _ 4 , (ij'a)j ; the former pair, owing to the near coincidence of 1 + m with c , having a very small divisor. The corresponding values of the coefficients in A are carried to more places and the slow progress of the approximations to the first pair was avoided as before. Section (viii). Characteristic a 2 . 61. Here t=o. Value of A. Coefficients of i. C n - 4 — -ooooo 01 3 — -OOOOI 02 2 -•00025 53 1 + •05977 63 -•03355 85 — 1 + •00220 05 — 2 — •01280 45 -3 — '00006 20 —4 j — -ooooo 04 Sum ... ! + ■01528 58 ii2 Mr. Ernest W. Brown, Theory of the The solution is Values of i. («% 4 + •00000 0007 3 + -ooooo 097 2 -f -00009 385 I + •00722 77 O — •00960 28 — I — •00720 42 — 2 —•00142 707 -3 — •00000 223 -4 — - ooooo 0008 Sum — -01091 38 These coefficients were obtained to six places of decimals on p. 157 of my paper referred to in § 50 above by using the homogeneous equations. They have been recalculated by the method of this chapter and extended to seven places. In comparing the earlier results with those given here, the transformation noted at the end of Sect, (ii) of this chapter must be made. The errors in the earlier results were very small. 62. Having finished the terms of the second order in u, we now come to those in z which are distinguished by having the first power of k in all their characteristics. For these terms the equations (s)-(8) of Sect, (i) of this chapter are used. The calculations cause very little trouble and are not long. In the short-period inequalities with small divisors, the progress of the approximations is not very slow. Long-period inequalities do not pro- duce small divisors. Section (ix). Characteristic ke, 63. Here 2«" + t takes the two sets of values ±(2& + g + c)and ±(2i + g-c), each set giving an independent set of equations of condition for the co- efficients. Motion of the Moon. Values of A. Coefficients of i. ftt+g + c f ,,- + g_e 5 + •ooooo 0001 + •ooooo OOIO 4 + •ooooo 0054 + •ooooo 0817 3 + 'OOOOO 4225 + •00006 2854 2 + •00032 0242 + •00398 8176 I + •0204s 4910 + •16874 9622 + •87305 5636 + •86734 1740 — I + •13526 5495 _ ■01316 9637 — 2 — •00250 6140 — •00046 2414 -3 •00008 9740 - •ooooo 7977 -4 - •ooooo 1579 - •ooooo 0106 -5 — •ooooo 0019 — 'ooooo 0001 Sum + i •02650 3085 + 1 ■02650 3084 : Coef. of £ 2i -s-°=_coef. of £-*+*+°, Coef. of f"-s+°=_ coef. of t !i4 n The solution is « ket =akeS,[(eA) j ^ + « + « + ( e '*')(^~ g " c +^).ff ! ' + *" c + (^')^" g * ] =akoS ( [(£&) ( (£* + » +e - t 2i ~ g - c ) + (e^M****-* - r 2 ^ g+0 )]. Values of i. (J<)i. (« 74-. 5 _L ooooo 0000 1 I i 4 + ooooo 00006 + ooooo 00149 3 i *> 1 + ooooo 00740 3 + ooooo 20814 9 2 + 00001 00784. 2 + 00030 77463 1 + 00146 61399 + 05543 3972 + 2 5°9 I 3591 _ 73687 7762 — 1 — 1 1999 3099 - 008 1 1 4895 \ —2 — 00x79 7078 - 00004 46815 ! " 3 - ooooo 93584 - ooooo 03014 9 ! -4 - ooooo 00626 5 — ooooo 00021 4 — c .J Sum - ooooo 00004 I — ooooo ooooo I ' + 13059 0286 — ■68929 3827 There is no short-period inequality with a small divisor. Royal Asteoh. Soc. Vol. LIII. Mr. Ernest W. Beown, Theory of the Section (x). Characteristic ke'. 64. Here 22' + t has the two sets of values ±(2i + g + m) and giviDg rise to two independent sets of equations of condition. Values of A. Coefficients of K 2i + g—m] i. /■SiH-g^-m^ f2i + g- m j 5 + 'OOOOO 0002 4 — -ooooo 0034 + 'OOOOO 0222 3 — •00000 2530 + •00001 7 26 3 2 — •00016 8839 + •00114 5257 1 -•00794 5172 + •05318 2849 — •00150 67489 — •00386 17596 — 1 + •05329 66967 -•00788 15823 — 2 — •00090 3017 + •00013 7147 -3 — 'OO002 6532 + 'OOOOO 39 2 3 -4 — -ooooo 0433 + 'OOOOO 0061 -5 — -ooooo 0006 + 'OOOOO OOOI Sum +•04274 3385 + •04274 3383 Coef. of ^"8~ m = -coef, of £-«+«+» Coef, of £»-«+»>=: -coef. of {~*+g~-<*. The solution is -£-2*-S-'») + (^'£) ( (£!>+g--m_£-a(--g+m)J_ Values of i. (#)»• (V'k)i. 5 + "OOOOO OOOOO 2 4 — -ooooo 00004 s + *ooooo 00030 2 3 — •ooooo 00549 5 + ooooo 03925 4 2 — •ooooo 71590 + •00005 16918 1 — •OOO92 26l 12 + •00680 83683 — -01600 9252 + •01924 4655 — 1 — "II002 672I + '04091 0818 — 2 — -00033 46147 + -00008 41 1 28 ! -3 — ■ooooo 20106 + -ooooo 04271 [ —4 — -ooooo 00139 3 + •00000 00026 7 -5 — 'OOOOO OOOO I + *ooooo ooooo I Sum — •12730 2438 + "06710 0471 Motion of the Moon. i 1 5 The short-period inequalities having small divisors are those with co- efficients (rjk) 0) (y]k)_i, (t]'k) , (V^)-i5 for these the corresponding values of A, are carried to ten places of decimals. Section (xi). Characteristic ka. 65. Here 2/ + T =±(2? 1 + g). The suffix of/, will be omitted. Value of A, Coefficients of 2i. C' 2(+g - - _ __. 9 7 5 3 1 1 ! _1 i ~ S i -7 ! - 9 + 'ooooo 0008 + ■00000 0534 + •00002 6082 — '00008 7162 — •19302 7070 --18559 984? + -00734 5267 + '00002 8303 — 'ooooo 0473 — *ooooo 00 1 3 Sum ■-•57131 6371 The solution is Coef. of £*-*=- coef. of t" li ^.. £^i=aka2 ; (ia) i (^'^-f-- i - 6 ;. zi otlll. ix6 Mr. Ernest TV, Brown, Theory of the Motion of the Norm, Values of j 2i. (&»),. I 9 + 'OOOOO ooooo ^ '• 7 + *ooooo 00072 1 5 4- - ooooo 05027 3 -■00005 33463 i — •06026 0507 — i 4- '15913 4186 -3 + •00375 4653 ! -s • 4- -ooooo 67585 : -7 4- -ooooo 00082 -9 — 'OOOOO ooooo 9 Sum 4--I0258 2262 ! There are no short-period Inequalities with small divisors, (To be continued,) J ppendix.— The terms of the third order (Chap, V.) are in process of calculation. The following results, properly belonging to Chap. V., have already been obtained and may ht recorded here. They have been used as the basis of a paper in the Monthly Kotie™, "Or, the Mean Motions of the Lunar Perigee and Node" (1897 March). Let e = e„ + e 2 c e , + e'"e i; „ + k 2 c k3 , g = g. + e 2 §V- + «'%,/= + k2 g k - Then c e _=+ -00268 575, g e ,= +-00318 579, %.,= --03465 53= g e --= +-00564 65, e^= +'05384 91, g t ,= --00806 633. The value of g k » is quoted from Mr. Co-well's paper referred to in § 31 above. Ilaverford College, Pa, U.S.A. : 1897 May 12. Theory of the Motion of the Moon; containing a New Calculation of the Expres- sions for the Coordinates of the Moon in Terms of the Time. By Ernest W. Brown, M.A., Sc.D., F.R.S. [Received 1899 February 3.] PART II. CHAPTER V. In the following pages I continue the Memoir the first part of which was published under the same title in the Memoirs of the Royal Astronomical Society in 1897. The general theory was given in Chap, I. as completely as I could then foresee would be necessary for the whole work. In Chaps. II., III., IY. the numerical results up to and inclusive of the terms of the second order were given. As the work progressed modifications tending to simplify or abbreviate the calculations naturally occurred. These, however, were fewer than might have been expected. The most important of them is given in Section (ii) below, consisting of a new method for finding the values of the final coefficients, after those of the quantities denoted in Chap. I. by A have been obtained. Previously this process consisted in solving, by continued approximation, for each characteristic and argument, a set of linear equations which were generally about 20 in number, with 20 unknowns. This process I had not succeeded in arranging conveniently for the computer, and as, in the terms of the third order, it involved about one-third of the whole work, some change was desirable. The investigation which led up to the new 164 Mr. Erxest AV. Browx, Theory of the method for tliis purpose was made from a different point of view some three years ago ; its usefulness became apparent directly the arrangement of the work for the computer was under consideration. Moreover, the numerical errors made in solving the linear equations were quite numerous ; under the method of Section (ii) they have been no more frequent than in other parts of the work. The numerical results given below are the values for all the terms of the third order, with certain subsidiary results which the above-mentioned modifi- cations require. The degree of accuracy which the theory up to this point attains may be best appreciated by a statement of the maximum number of coefficients of the fourth and higher orders, which may be as great as 1" of arc in longitude. Of the fourth order, one of 4" and three of 2", which contain only e, k in their characteristics, and twenty of 1" ; of the fifth order there are two of 1", which involve both e, k. Moreover, the principal parts of nearly all. of these are the purely elliptic terms. I have received very great help in performing the calculations from Mr. Ira. I. Sterner, A.B., of Haverford College, who has, since 1897 Septem- ber, been my only assistant. That so much has been achieved in the time we were able to give is largely due to his accuracy and capability. Much of the work done by him would scarcely have been attempted by an ordinary computer without very extended instructions, while his knowledge . of arithmetical processes has not only been a great saving of time and labour, but has made the chief part of my task — that of testing and correcting his work — a comparatively light one.* The following is the table of contents of Chap. V. : — Section (i). A brief outline of the application of the general method to the terms of the third order in the calculation of the series A. Section (ii). JS T ew method for solving the linear equations when the series A have been obtained. Numerical values of certain quantities required in this method. Section (iii). Modification of this method, in order to avoid, as far as possible, the loss of accuracy arising with long-period terms. Section (iv). The method of calculating the new parts of the motions of * A portion of the expense of making the computations necessary to obtain the results given below has been met by a grant from the Government Grant Fund of the Royal Society. Motion of the Moon, ^5 the perigee and node, and the coefficients arising therewith. Numerical values of certain quantities required. Section (v). The final numerical results for the series A, and for the coefficients of all terms of the third order in v. :.* Section (i). Formula' and 7 est*. 66. About two-thirds of the whole labour of obtaining the coefficients of the third order consists in the calculation of A. The products in the third line of equation (17) and of the second line of (18) in Chap. I., are formed by putting and choosing out the parts of the third order. The products in the fourth and fifth lines of (17) and the third line of (18) are obtained with Mfi = W<l, s p— s l> % — ^1 Here ll^lle + U^ + Ua, »i=%, U.f= U e , + U ee , + U e ,~. + U w + W e . -f U lU + It,,, the expression for s 2 being similar to that for n 2 . The parts arising from 8 1} namely.-— 1 £-\ - $° Q \ are treated in like os dz manner. The parts arising from the first terms in the right-hand members of the equations, are treated in Section (iv) below ; they only appear when the terms of arguments 2i±c, 21 ±g are under consideration. The general method of procedure has been as follows : — The computer having performed the calculations allotted to him, I go over them all and test them by all the means available. Each multiplication of series is tested by sums as explained in Section (viii). Chap. I. The final values of A are tested in the same manner, but in larger groups, so as to make certain that no series of terms has been omitted ; for this, the values of A for each character- * It is intended to give the subsidiary results in a final chapter, or chapters, when tlio whole theory has been completed. Royal Astros. Soc, Vol. LIII. a a ra 1 66 Mr. Ernest W. Brown, Theory of the istic, with £= i , are all added together and the sums compared with the corresponding sums obtained as directly as possible from the algebraical formulas, which become quite simple when £= i . It is true that this method will not test for all kinds of errors, e.g. the accidental interchange of u and s, but reliance has been placed less on test equations than on the care which has been taken to avoid errors. The calculations made after the method of Section (ii) were treated in the same way as far as possible ; a further and very searching test was obtained by forming Dh/ M Dn A with ?=i, for each value of r and substituting the results in equation (24) of Chap. I. In the cases where calculations were not turned over to the computer, they were gone over again after an interval of time and, when possible, tested. Increased accuracy in making the calculations was always found whenever blocks of the same nature were performed together. Section (ii). Method of solving the Linear Differential Equations. 67. The equation (24) of Chap. I. to be solved is (D + myu x + Mu x +A r s x c, i -a.XAi:=A' ... . (1) Let Then u/, s/, j=i, 2, 3, form three particular integrals of (1) when .4=0. In a paper to be published elsewhere, I have shown that the fourth particular integral can be given the forms t l =u/=u 3 ' J D- 1 ( --t-,- 1 V u 3 s/ / =Iu.' r S -L U2 -_ u >' s A + 73-i f Ca s/uZ-u.V / m , D\i a ' _DaA\ m l 2 ° L <s/ i U 3 'S 3 ' U 3 'S 3 ' V U/ S//IJ «=s 4 '=u/, where C 12 is a constant given by C K =s 2 'i)u/-u 1 'Z)s/+u 2 'i)s 1 '-s/i)ii/-2m(s 1 'u./-ti l 's 2 ') = 25(21 + 1 +m + c ) 2 i + 23(2* - 1 - m + c )f_i' 2 , the bar over u/ denoting that %~ x has been put for £ that is -< for t in the expression for u/. Motion of the Moon. iQy It is further shown that C -=u 3 'i)- 1 q + u^=U3' t (M-M')(<-gq + u i ; ) where q is the constant term uncfer the sign D l in the expansion of u 4 70 12 in powers of ^ and u 4 £ is a series of the same form as u 3 . Finally, it is shown that the solution of (i) when A is not zero can be put into the form Mi= tJ7 2 [ u i' Z) ~ I ( s *'^'+ Us'i'O-Uj'D-'foM' + u,'!')] + u 4 ;Z>~- 1 (s 3 M' + u 3 'I') -u.'n-ms^A' + n.UO-qD-^sW+uM')} (2) in which the bar over A has the same meaning as before. This is the form required ; it will be noticed at once that most of th operations will consist of multiplications of series, and will therefore be h line with the work which was necessary for the calculation of A. 68. It is advisable to make a few changes in order that the formula may be more compact. We first observe that for all purposes except the parts due to S3 , we require, not u„ but u x /u ; the parts due to £3 have the factor m 2 , and are therefore small, and the multiplication of u x /u by u is easier than that of u % by i/u owing to the fact that u v has one coefficient unity. As there is no increase of trouble involved in finding uju m we shall do so. Write therefore — Uj , u 2 8l s,. (3) *o *o £q *q U 3 =U 3 'f- 1 , 85 = 83'^ (4) tt _af V^ o<> T t _ae 'u 2 '4' c ' > TT _au 3 ' TT au 4 ; , . These ten quantities are then all series of the form 2 jt? ( £ 2i with numerical coefficients, and the constant coefficients in u 2 , s 3 , U 3 , are all unity. Also, D- 1 (X£°) may be written £° (D + c)-*X; the operation D^ % consisting of a division by t, the operation (D + c)~ l £ T will consist of a division by t + c. The formula (2) may now be written X ^=U l (2)-c )- 1 (s 2 ,i + ii 2 Z)-U 2 (D + c )-\s l A+u l A) + -U i I)- 1 (s 3 A+u 3 A) -U 3 /)- 1 {(M + u l I)- q _Z)^ ( S3^+u 3 Z)!- ( 6 ) 1 68 Mr. Ernest W. Brown, Theory of the Next, put Then since W>= Si A + u 4 J, T^D^'qY.-W,) S,— U,, 11;— S 2 , u 3 — -S 3 , U 4 = s 4l D——D, (7) and therefore -(D + Co^Xs^l + UjJj^Q,, UjJ"=-M. u ! .r=s7I7 V^V,, TY=W„ T»=-T„ the equation (6) takes the final form 1 ^•=TJ 1 Q x + UA + TJ 3 T i + U 4 V !l (8) J 69. The calculations are arranged as follows : — The series u 2 , s 2) s 3; s 4 , Ux, L T 3 , U 3 , U 4 are obtained once for all by multiplication or " special values," and the logarithms of the coefficients written out on slips, as explained in Chap. I., Section (viii). The slips containing A having been made out for each value of \ the multiplications s. 2 A, u.,i, s s A, s 4 i (9) are performed by the computer, and thence the values of Q A , Y A , T A are, by a few easy processes, obtained. The slips containing these latter are then made out, those for Q A serving for Q A . Finally the multiplications U,Q« U 2 Q-„ U 3 T„ u 4 V> ... ... (10) are performed by the computer, and thence, by addition, the value of uju is obtained. The series A are of the form 2wp,£ 2i+T + £ (j p,-£ w ~ 7 . When t is not zero, the proc is for each such double set involves sixteen multiplications of series of tl : form (Sp^ 2 ')^^ 2 '). When t=o, half the number suffices. When r = c,; a new part of the motion of the perigee is under consideration, and though the same process may be used, it is not convenient (see Section (iv) below). When t=o and 21 is even, it would appear that T A gives rise to terms with the time as a factor. It is shown in the paper referred to at the beginning of this section that such terms can always be made to disappear. However, none such occur amongst the terms of the third order. In any Motion of the Moon. 169 case the method of this section will not be used for them, as the approxima- tions in the ordinary method are rapid, owing to the absence of any small divisors. The same method may be applied, using well-known formulae, to the calculation of z k . but it appeared easier, at any rate in the terms of the third order, to use the method given in Chap, I. The chief objection to, the method consists in the fact that the small coefficients which accompany large values of i appear as differences between comparatively large 'numbers. This fact does not impair the required accuracy of the results, but the multiplications are much longer than they otherwise would be. Thus the coefficient of ^ 6 ~ 3 '" in u e .-./e 3 t/ appears as follows : — ■ From UjQ, — '00006 418 ,1 VzQn +'OOOo6 496 „ U 3 Tj + 'ooooo 316 » "C 4 V, — -ooooo 39 2 Sum + 'ooooo 002 On the other hand, as the presence of this difficulty can be shown to be peculiar to the method, it furnishes a means of detecting a certain class of error. 70. The following are the numerical results required for the method of this Section : — Values of the coefficients of £ 2 ' in i s,,. ■a,,. %■ s v 4 — 'OOOOO 002 3 — '00000 017 — 'OOOOO 279 + -ooooo 0012 — 'ooooo 003 i 2 + '00006 960 — -00041 211 + -ooooo 0491 — 'OOOOO 147 1 1 + ■19895 557 — -07424 980 — -00869 5747 + •03592 927 i -•33619 118 + 1 — I -•93146 358 1 — 1 — '00196 357 + -00016 930 - -00454 7122 — '00648 I48 ! — 2 — 'OOOOI 349 — -ooooo 090 — -00002 9393 — -00004 410 j -3 — '00000 010 — -ooooo 001 — -ooooo 0210 — 'OOOOO 032 ! —4 — -ooooo 0002 Sum - -13914 334 + "9255° 367 — 1-01327 1971 —-•90206 171 • 170 Mi\ Ernest W. Brown, Theory of the Values of the Coefficients of £ z * in i U,. u 2 . U,. u. 4 — 'OOOOO 003 3 — "OOOOO 015 — 'OOOOO 417 + 'OOOOO 013 — '00000 022 2 — •00002 034 — '00066 197 + - OO0OI 892 — "00003 094 1 —•00317 658 — '16544 214 + -00303 148 — •00506 996 -•73476 98s + 2-18209 t-5 + 1 -•93154 984 — 1 + ■42803 606 + -01934 433 + -01739 126 + •02782 838 — 2 +•00387 413 + 'ooo 1 6 59° + "00015 057 + -00024 067 -3 + '00003 326 + 'OOOOO 142 + 'OOOOO 129 + -ooooo 206 -4 + 'OOOOO 029 + 'OOOOO 001 + 'OOOOO 001 + "OOOOO 002 Sum — •30602 318 + 2-03549 49 + 1*02059 366 -•90857 983 q= + i-5i4i5 29. 71. In the terms of the third order the above process has been actually used. In those of higher orders a slight abbreviation of the work will be made by the following change. Put A _ 3 M *0-> A Po (10). and, instead of u 2 , s 2 , s 3 , s 4 use each of these series multiplied by Zw<£~ l /4pi. The latter four new series can be obtained once for all. The advantage gained from this change will be seen by looking at equation (17) of Chap. I. The terms A consist of two parts. The first is the part due to the first line of the second member of the equation ; the terms in this line are small, being due to Sc and £3, and their multiplication by APo/ZW-g- 1 to get the corresponding part of A 1 will be short. The second is the part due to the succeeding lines of the equation ; these always form the principal part of A, and they all have the above-mentioned factor. Section (iii). The Terms of Long Period. 72. Small divisors arise when the period of any term approximates to that of the principal elliptic term or when it is long. In the former case, the small divisor arises in the first and second terras of (8), being due to the Motion of the Moon. 171 operators (D±c )~ 1 , In the latter case, as was pointed out (Chap. I., § 28 (c)), the square of the divisor occurs, but this difficulty may be avoided by the use of the homogeneous equation (21). I shall now show how the latter equation may be adapted when the methods of the last section are used. The long-period small divisor arises only in V, T, owing to the operator D~ x . But (omitting the suffix X for brevity) T=Z>-'(qV-W), Y-=D~i(As, + Au 3 ), so that the square of the small divisor arises in the part D 1 (qV)=qD--(As 3 + Au s ) of T. This is therefore the term to be considered. We have BY=As 3 + Iu 3 =l(AiDs + Ai;- l Du ) =l{(A>:-hi, ti -A'Cs^ + Ai:--\Du ii -u a )-\-A2(Ds (l + s )) ... (11) But by equation (21), Chap. I., u D\ + u x D\—s D-u x — s x D :i u t) ~2mD (s u x + u 6 s x ) + ^vo?(u a u x — s u s x ) = a 2 A. A ', where A' denotes all the terms of characteristic a except those due to u K , s M in the expansion of D(sI)u-uDs + 2mus) + 3 m%s i -u i ) + s^ -J*® 1 txz) 2 as On x ' This equation may be written — or, using equation (24) of Chap. I., u A£ - 1 — s a Ai=an\'. Whence, restoring the suffix a, Y x =D-K\ x ' + I D-^{A^\Du a -i k ) + A-C(Ds + s )} (13) (X where a 2 X I)"" 1 A/ will denote the known terms of characteristic X, in sDu-uBs+2raus + .D' l \ 3 m%s i -u i ) + s^-ui^) (14) 1 2 ' Os Cu ) ' rm 172 Mr. Erxest ~W, Brown, Theory of the The portion of (14) under the operator D^ 1 contains the factor m 2 at least ; hence the effect of the small divisor (which is never of an order higher than m 2 ) is neutralised in the first term of V. The same thing occurs with the remainder of the expression for V. owing to the fact that J)u —u and D$ +s contain the factor m 3 at least. Hence V can he found to the same degree of accuracy as A, and the loss of accuracy in u k is limited to that due to the first power of the small divisor. In using this we first find ^1 as usual, and with it calculate the coefficient of the particular power of ~ in the second part of (13). The terms contain- ing this power of £ for the given value of ?. are then chosen out of the expression (14), and thence V is obtained to the recpiired accuracy for this power of %. The method of Section (ii) serves for all the other powers of £. Section (iv). The New Part* of the Motions of the Perigee and Node. 73. The method of calculating c A/e is explained in Chap. I., § 28 (b) ; when this quantity has been obtained A is completely known. Putting X = X ; (§25) and omitting one of the equations for i=o, we can solve the linear equations by continued approximation. The omitted equation serves as a test. Either this or the method of Section (ii) somewhat modified, can be used, but it is more convenient to proceed in the following manner. In § 28 (b), we have put A = B + c Ah b. Hence write the solution in the form v^v+XoC/'x+ov.co,, J ' '" "* 1IS) except for i=o. Here ^ jX/, are obtained by solving equation (26) of Chap. I. with A i =B i ,A[=B',-K Q =o=^, The terms {f\ 7i , (f\ X are obtained by solving the same equation with A i =o=A i f , x =A ' (§25), in terms of A . (If, instead of x =?. / , we had put ^o—^o'— 1 ) we should have found s,., s/.) The terms (c),-, (c')i are obtained by solving with Ai—b i} J./ = 6/, A =O=A . In all three cases the equations for i=o are omitted. Motion of the Moon. ^73 We now substitute the values of \ h x' ft thus obtained, in the two equations for i=o and obtain two equations of the form i*'o + Wo') + <Vo(eo')=o 1 (l6) in which everything is known except h . If the work be correct these should give the same value of A 0) and thence, substituting in (15), the values of A,, x' . The coefficients (/),-, (/),■, (c) t , (c') t being independent of x, are found once for all ; they are given below. The advantage of this procedure arises from the omission of the equations for i=o in the solution of the linear equations : the approximations are rapid. The same process is used for the nodal motion with the c-equation. But as we here put x =o = /\ ' (§ 26). the equations (15), (16) reduce to h=i\ + gx*(ff);, (except i=o) l\ + gxlk($/o) = ••• • (is)' (16)' The equation (16)' serves then as a test. In all coefficients arising in the --equation we have q i =—q'^ ; . 74. The following are the numerical results required for the method of this Section : — •■. (/);• (f% 3 — •00000 009 — .00000 039 2 — 'ooooi 269 — •00006 201 1 — •00213 J86 — '01412 29 — T + '01261 31 — •00293 22 4 — 2 + '00002 655 — 'ooooo 071 —3 — •00000 001 — '01711 83 Sum . . . + •01049 S° (/o) =+6-97857 932 (/'o)= + 2*34613 680 (e ) = + 1-07940 266 (c'o) =+ .02014 43 8 Royal Astros. Soc. Vol. LIII. B B 174 Mr. Ernest W. Brown, Theory of the i. (0)i. (0% (3)i- 4 — "ooooo 007 3 — 'OOOOO 002 — -ooooo 936 - •ooooo 001 2 — 'ooooo 348 — '00135 001 — •ooooo 298 I O — I — -00071 843 - -22311 94 — •ooiii 579 + -71453 80 + -00038 928 + •I9835 12 — 2 -j- -ooooi 970 4- -ooooo 037 + ■00030 486 -3 — 'OOOOO 066 + •ooooo 118 -4 — -ooooo 001 + •ooooo 001 Sum + 71383 Si — -22408 92 + •19753 8S Section (v). Values of A, u& YaX, i^/aA. 75* The following tables show the characteristics, arguments, and types of coefficients of the terms of the third order according to the scheme adopted in Chap. I., Section (iv). The numerical results are given below in the same order, and will be found in the §§ given in the first columns. «' A Arguments. Types of Coefficients in %f-'/aA. 76 e 3 2i±3C ; 2i + c (*'), (*") ; (A'), ( f£ ' 2 ) 77 eV 2i±(2c + m) ; 2i±(20 — m) ; 2i±m (S-n), (e'V) ; (eW), («".») ■ («'„), («v) 78 ee' 2 2J + (c+.2m); 2t±(c— 2m); 2'i±0 ( £J? 2 ), ( t y 2 ) ; ( £I / 2 ), ( t y) ; ( £Jp? '), (,' m ') 79 e /3 2 ^±3 m > 2i±m (v 3 ), W 3 ) ', (vW), inn' % ) 80 ek 2 2i + (o + 2g) ; 2i + (c — 2g) ; 2i + {*&), (e'F 2 ) ; (ek'*), (t'¥) ; (tkkf), (e'kk') 81 e'k 2 2i±(m + 2g); 2-i±(m — 2g); 2i + m (nk% {n'V*) ■ (r,k'*), (,'A*) ; („**'), (,/kk*) 82 e 2 a 2i { dz2C ; 2-ij (A), (e' 2 «) ; («'„) 83 ee'a 2i 1 ±(c4-m); 2-i 1 + : (c — m) (eijct), (t'j/a) ; ( f ij'a), ( £ 'qa) 84 e'' 2 a 2'i 1 ±2m ; 21 y (?/ 2 a), (ij' 2 a) ; (ijj/'«) 85 k 2 a 2l,±2g; 2%y (F«), (k' 9 -a) ; (kk'a) 86 ea 2 2i + ( f a 2 ), (,V) 87 / e or 2i±m (ij.i 2 ), (ij'a 2 ) 88 0. 21 x (a') Motion of the Moon. 175 § A Arguments. Types of Coefficients in f^/aA. 89 ke 2 ±( 2 i + g±2c); ±(2i + g) ±(kS-), ±{kt>*); ±(i«') 90 kee' ±{2i + g±(c + m)} ; ±{2i + g±(c-m)} + (&*), ± (W); ±(^V), ± (&'*) 91 ke' 2 + (2i + g±2m); +(2* + g) + (*,»), ±(*'/»); + (*„/) 92 k 3 ±(2i + 3 g); ±(2» + g) + (*»); ±(£W) 93 kea ±(2'h+g ± c) ±(&£a), ±(&£'a) 94 ke'a ±(2*i+g+m) ±(^ a )' ±(^'i' a ) 95 ka 2 ±(2i+g) ±(&a 2 ) The following long-period terms have been obtained with the required accuracy by the method of Section (iii) : — Arguments. ±( — 2 + 2C + m), ±(— 2 + 2C — m), ±m, ±( — 1 +c — m), Coefficients. («'„)„, («V)» (tlj'oi)-!, (fV) } The values of V x for these terms, obtained as shown in Section (ii), agreed, as far as they went, with the values obtained by Section (iii) ; this agreement furnished a valuable test. Equation (8) of Chap. I. was used to obtain (W)^ with sufficient accuracy. Preliminary values of the parts of the motion of the perigee and node having e 2 , e'% k 2 , as factors were given in an Appendix at the end of Chap. IV. The values found below differ slightly from these. This is partly due to the fact that they have been re-calculated by a different and more accurate method. In one case — that of g e = — an error was found in one of the final steps of the early calculations (where no test equation had been computed) ; this induced an error in c k , which was deduced from it by using the con- necting relation which I gave in a paper, "Investigations in the Lunar Theory."* These two quantities having been recalculated independently, and the values satisfying the relation just mentioned, they may be accepted as final. * Amer. Jour, Math., Vol. xvii. p. 349. i/ 6 Mr. Ernest W. Brown, Theory of the The numerical results now follow. The values of A K for the terms arising in u are given in two parts— those arising from the expansion of m%- T jr\ denoted by K K , and those arising from - a ® ' £-, denoted by 8 A . Then except for the exponents 21 ±c, 2^ ±g, where The numbers are the coefficients corresponding to the power of % (that is, the argument) which is placed at the head of each pair of columns. The separation of A k , B x in z is unnecessary, as the parts arising from 8 require but little calculation. The suffix of i x is omitted in the tables. Further details concerning the results will be found in Chaps. I.- IT. Motion of the Moon. 177 76. Characteristic e 3 . g , = o. Values of R e > = - -tie- K e „ = = B e ,. i. 2i + 3=- ii - 3c 21 +C. 2% — c. 5 + •00000 615 + '00000 OI5 + '00000 175 4 + 00000 025 + •00025 °37 + •00000 883 + '00009 *39 3 + 0000 1 5 01 + •00618 376 + •00046 189 + •00371 182 2 + 00079 199 + ■00932 740 + ■01879 136 + •08978 705 1 + °3 3 5 I 1864 -•05285 4356 + •45337 5630 + '00093 o745 + 79860 3673 + •09945 4882 -•29234 3773 -•09815 4555 — 1 — 18057 2 435 + •00264 8901 — •00031 9004 + "°5 8 5 1 8714 — 2 + 00415 328 + •00004 93 x + ■01207 313 + •00155 383 -3 + 00086 99i + •00000 074 + •00031 268 + -00002 902 -4 + 00002 152 + •00000 579 + 'OOOOO 048 -s + 00000 040 + 'ooooo 007 ' Sum + 65639 546 + •06506 716 + •19236 675 +•05647 029 Values of i. (Or (*"> ( 6V )>- («% 5 + •00000 010 + "OOOOO 002 4 + 'OOOOO 701 + - ooooo 009 + 'OOOOO 138 3 + '00000 014 + '00038 889 + •00000 678 + '00009 907 2 + ■00001 122 + •00176 65 + •00048 105 + •00528 276 1 + •00079 516 — ■00218 478 + •02520 232 — ■00568 84 + ■04147 214 + •00522 993 -•04231 894 -•04231 894 — 1 -•03023 295 + '00004 365 + •01685 68 + •00328 278 — 2 + •00184 26 + 'OOOOO 052 + •00068 577 + '00002 537 -3 + •00004 5°9 + 'OOOOO 001 + •00000 497 + 'OOOOO 031 -4 — 5 + 'OOOOO 032 + -ooooo 006 Sum + •01393 37 + •00525 18 + •00091 89 -•03931 56 j C e a = + '00268 571 178 Mr. Ernest ^Y. Browx, Theory of the J J. Characteristic eV. Values of K eV + S e v i. 21+2C-1 -m. 2i — 2e — m. 5 + "OOOOO 711 4 — •00000 O44 + •00033 841 3 — '00002 526 + •01192 768 — 'OOOOO 024 .2 - '00120 655 — 'OOOOO 006 + •21675 309 — •OOO02 IOO 1 - •04218 737 — 'OOOOO 553 2 — '2II26 on — •00007 0187 - 74778 991 — '00046 3*58 — -13819 518 — •00047 54" — 1 — •27045 446 — •00077 1829 - '00445 874 + '00046 0625 — 2 • + •04960 312 -•00054 348 — '00009 192 + 'OOOOO SSi -3 + •00141 627 — •00014 .708 — "OOOOO 152 +.•00000 006 -4 + ■00002 806 — •00000 166 — "OOOOO 003 -s + •00000 046 — 'OOOOO OC2 Sum — ] •01061 608 -•00193 282 — •12498 121 — -oooio C64 i. 2* + 2C - -m. 24- -2c + m. 5 + 'OOOOO 004 — 'OOOOO 108 4 + 'OOOOO 249 — "00005 063 3 + •00013 421 — '00181 981 — 'OOOOO 024 2 + •00580 739 — 'OOOOO 006 -•03489 248 — '00002 107 I + •16168 213 — 'OOOOO 5509 -•01655 646 — ■00012 2352 + •91682 513. — "00046 0625 + •18214 920 + •00208 0639 — I — •02119 681 + •00047 54" + •01811 441 — "00322 6913 — 2 — •00724 120 + •00007 019 + '00046 244 — •00003 856 -3 — •00020 x 93 + '00002 IOO + 'OOOOO 837 — "OOOOO 040 -4 — 'OOOOO 395 + •00000 024 + 'OOOOO 013 -5 — 'OOOOO 009 Sum + 1 •05580 741 + •00010 065 + ■14741 409 — •00132 890 Motion of the Moon. 170 Characteristic eV. Values of K eV + Q e v i. 21 + m. 21 — m. 5 — 'OOOOO 020 + -ooooo 109 4 — •0000 1 021 + -00006 185 3 — '00049 120 — 'OOOOO 002 + "00284 107 — 'OOOOO 002 2 -•01737 829 — 'OOOOO 227 + -09173 506 — 'OOOOO 226 1 — •32001 82 2 — '000 1 9 2836 + 1-28586 471 — -ooo 1 9 1967 — -11424 8ll + '00014 0471 — -10877 253 + -00072 573° — 1 + •27689 127 + -00449 4856 — -06274 5 2 7 — -00072 5730 — 2 + •01060 170 -•00134 464 - -00188 121 + .00019 197 -3 + •00023 060 — 'OOOO I 580 — -00003 788 + -ooooo 226 —4 + "OOOOO 392 — -ooooo 016 — 'OOOOO 057 + '00000 002 -s + 'OOOOO O06 — -ooooo 001 Sum — -16441 868 + '00307 960 + 1-20706 631 Values of i. (e'l),. («'Y >.■ («V) ( . (A> 5 + -ooooo 008 — 'OOOOO 001 4 + 'OOOOO 691 + -ooooo 002 — 'OOOOO 099 3 — 'OOOOO 029 + ■00047 850 + -ooooo 156 — '00006 734 2 — '00002 191 + ■02396 981 + •000 1 1 095 —■00327 SOS 1 -•00143 289 + •03768 72 + ■00590 206 + '02902 28 -•06288 748 — -01411 942 + •08337 053 + •02431 890 — 1 — -14741 25 — -00006 809 -■03817 68 + •00036 494 2 + •00695 O47 — '00000 TI2 — '00113 005 + 'OOOOO 631 3 + '00001 7 r 4 — -ooooo 002 — 'OOOOO 207 + 'OOOOO 007 —4 5 + 'OOOOO 036 — 'OOOOO 004 ] Sum — •20478 7i + •04795 39 + •05007 62 + •05037 16 150 Mr. EievEST ~W. Brows, Theory of the Characteristic eV, Values of i. ("V'- («Y)i. i 1 5 + •ooooo 001 I 4 — -00000 012 + •ooooo 076 3 — -ooooo 929 + •00005 597 2 — -00062 354 4 •00347 999 I - '02957 776 5 •12638 o33 O - -19519 75 4 •14774 72 — I + '04554 004 — ■00648 982 — 2 + -00017 353 — •00002 5°4 -3 + -ooooo 307 _ 'OOOOO 046 -4 + -ooooo 003 - •00000 001 -s 1 Sum ... — -17969 !S 4 •27115 1 39 i 78. Characteristic ee' 2 . Values of K e ,„ + Q ee „. i. 2*' + e- 2m. 21- -c — 2m. 5 + •ooooo 241 4 + •ooooo 017 + •00012 est 3 + •ooooo 873 + ■o°5 1 3 044 — 'OOOOO 014 2 + •00029 198 + "OOOOO 001 + •12386 6 59 — 'OOOOI 320 1 + •00371 387 — 'OOOOO 380 + •09957 946 — '00 1 20 073 — •ioioi 661 — -00138 S°3 - •03198 329 + -00293 3 2 9 — i + ■05016 537 + -06034 441 + •00053 064 — -00045 811 — 2 + •02162 607 — '01016 166 + •00002 797 — -ooooo 70s "3 + '00056 850 — "00010 231 + •ooooo 062 — -ooooo 008 1 -4 + ■0000 1 °33 — 'OOOOO °99 + •ooooo 001 ~5 Sum . + ■ooooo 018 — -ooooo 001 136 + -00125 398 _ •02463 141 + '04869 062 + •19728 Motion of the Moon. 181 Characteristic ee' 2 . Values of K ee ,i + 8 K „. i. 2(*+C — 2m. 2J- - c + 2m. 5 + •ooooo 016 + •ooooo 006 4 + ■ooooi 003 + •ooooo 193 3 + •00049 475 + ■00006 588 2 + •01857 417 - •ooooo °53 + •00106 031 — 'OOOOO 069 I + •38487 310 — •00004 6i5 - '01398 063 — '00028 638 O + •22941 999 •00247 794 + •08802 983 + -01764 269 — I — •00540 404 - •00018 882 + ■06570 344 - "02534 819 — 2 + •00015 577 •000 1 1 935 + •00202 853 - -00037 010 -3 + •ooooo 637 •ooooo 158 + •00003 953 — 'OOOOO 403 -4 + •ooooo 014 — •ooooo 002 + •ooooo 063 — "OOOOO 004 -5 + ■ooooo 001 Sum + •62813 044 - •00283 439 + •14294 95° — '00836 674 i. 5 2i + 0. 22- -c. — 'OOOOO 005 — 'OOOOO 073 4 — -ooooo 310 — '00003 719 3 — -00016 037 - '00153 051 — 'OOOOO Oil 2 — -00628 749 — 'OOOOO 043 - -03870 578 — 'OOOOI 092 1 — -14287 981 — '00003 572 - -14727 365 — '00093 229 - '17975 583 - -00139 125 — -06087 008 4- '01360 no — 1 - '05798 850 — '00761 209 - '02253 804 + '01 000 455 — 2 — -00621 038 + -00299 069 — '00064 816 ■f '00012 062 - "3 — '000 1 5 985 + 'O0OO2 956 — 'OOOOI 220 + 'OOOOO 126 -4 — 'OOOOO 283 + 'OOOOO 029 — -ooooo 020 + 'OOOOO 001 -5 Sum ... — -ooooo 005 - '39344 826 — -00601 895 — '27161 654 + '02278 422 Royal Astrox. Soc, Vol. LIII. c c 182 Mr. Ernest W. Brown, Theory of the Characteristic ee /2 . Values of i. («?% 1 («V)«. (*V"-)i- (*'v% 4 + •ooooo 210 + •ooooo 010 3 + ■ooooo on + •00015 848 + •ooooo 759 — 'OOOOO 227 2 + •ooooo 579 + •00994 322 + •00050 707 — '0005 1 717 I — •ooooo 203 + •36932 90 + •02381 252 — 'IOII2 9i O — •05024 49 + •10676 52 + •14026 79 — '23325 20 — I — •78467 7o — •0000 1 322 + •29493 28 — "00039 221 — 2 — •00147 012 + •ooooo 026 + •00003 6 55 4- "00003 448 -3 + •ooooo 949 + •ooooo 001 — •ooooo 018 + 'OOOOO 047 -4 + •ooooo 012 r Sum ... — •83637 85 + •48618 51 + •45956 44 - "33525 78 i. («?i')i- («W)i. 4 - •00000 003 — 'ooooo °59 3 — •ooooo 236 — -00004 352 2 — ■00016 176 — -00264 878 1 - •00797 920 - -08755 90 — •02023 661 — -02023 661 — 1 + •14228 77 + '00040 089 — 2 + ■00037 254 — "OOOOO 957 -3 — •ooooo 231 — 'OOOOO 014 -4 — 'ooooo 003 Sum + •11427 79 — '11009 73 • •03465 60 Motion of the Moon. 183 79. Characteristic e n . Values of K e -, + Q ^,. i. 2i+3tn. 21- -3m. 5 + •00000 026 4 — -ooooo 001 + •00001 439 3 + 'OOOOO 003 + •00063 617 — -ooooo OOI 2 + •00002 943 — "OOOOO 001 + •01785 243 — -ooooo 180 i + •00165 270 — -ooooo 866 + ■04256 833 — ■00017 494 + •02364 0522 -•00875 9209 + '01902 8519 — ■01242 3782 — 1 + •00827 230 — •18242 850 + •00027 208 — •00041 049 — 2 + •00292 678 -•00159 328 + 'OOOOO 321 — •ooooo 047 -3 + -00006 929 — •0000 1 486 + 'OOOOO OOI -4 + -ooooo 116 — 'OOOOO 014 -5 _ Sum + - ooooo 002 + '03 6 59 222 — •19280 466 + •08037 539 — •01301 148 i. 2i + m. 2s — m. s + 'OOOOO OOI — "OOOOO 010 4 + -ooooo 066 — -ooooo 629 3 + -00002 869 — '00028 158 — '00000 OOI 2 + •00072 679 — 'OOOOO OOI — •00825 586 — 'OOOOO 123 1 + •00088 954 — •00002 564 — '04906 880 — '0001 1 396 — •02263 5731 -•00385 1214 — '02019 3073 -•00275 2931 — 1 — -00919 324 + •08674 561 + '00009 023 — ■00087 6 33 — 2 — •00124 392 + •00067 905 + •00010 367 — '00005 386 -3 — -00002 899 + 'OOOOO 633 + 'OOOOO 278 — 'OOOOO 064 -4 ~5 — -ooooo °5i + 'OOOOO 005 + 'OOOOO 002 Sum -•03145 670 +■08355 418 — •07760 900 -•00379 896 1 84 Mr. Ernest AY. Brown, Theory of th ie Characteristic e' 3 . Values of i. (l 3 );. (i")« (rf-n')i (vn' 2 )i- 4 4- 'ooooo 02I + "OOOOO 001 — "OOOOO 008 3 + •00001 620 + 'OOOOO 056 — •ooooo 65s 2 — 'OOOOO 004 + •00113 404 + '00002 762 — "00044 497 I — '00003 123 + •05909 Il8 + "00021 277 —•02145 800 O -■04032 334 + •06189 440 — •00900 66 — •00500 46 — I —•26598 707 — '00002 034 + •08932 148 — '00064 95° *— 2 + •00004 936 — 'OOOOI 321 + 'OOOOO °35 "3 + 'OOOOO 132 — •ooooo 049 + 'OOOOO 003 —4 4- 'ooooo 001 1 Sum -•30629 099 + ■12211 569 + •08054 21 —•02756 33 80. Characteristic ek 2 . Sek a = °- Values of K A *=A ei *. + 2(— C+ 2g. 'OOOOO 001 + 'OOOOO 066 + •00004 388 + •00236 501 + •°7579 4392 — I '337" 7389 J_ • 5938 0541 + •00187 632 + •00003 410 + 'OOOOO 049 .— I '19762 199 Motion of the Moon. 185 Characteristic ek ii^—o. Values of 7f ck „=i? 6k =. ?'. 2/ + e. 21 — C. 5 — 'OOOOO 005 4 — 'O000O 025 — 'OOOOO 3 r 3 3 — •0000 1 523 — •00015 379 2 — "00072 653 —•00524 052 1 -•02123 4090 -•09199 9007 -•03507 4881: — '02732 4124 — 1 -•04932 6078 + '06866 6563 — 2 + •01296 733 + •00145 824 -3 + '00028 345 + ■00002 175 -4 + 'OOOOO 425 + 'OOOOO 028 -5 + 'OOOOO 007 Sum -•09312 201 -•05457 379 Values of 4 3 2 1 o — 1 — 2 3 -4 Sum (.#% + •00000 027 + •00002 256 + •00138 166 — -07120 080 + •00889 21 + •00005 33t + -ooooo 030 — ■06085 06 (.'*«),. + •00000 053 + •00004 989 + •003? }6 -•01711 531 + •24987 870 + •00146 418 + •00001 008 + -ooooo 007 + •23814 17 (<*' 2 X- («'**){■ + •00000 002 + -ooooo 085 — 'OOOOO 008 i — 'OOOOI 633 — "OOOOI 355 i -•02889 967 -•00235 066 + ■64704 18 -•46343 59 +•05538 425 + •17077 937 ! + '00030 609 + •00029 727 ' + 'OOOOO 207 + 'OOOOO 136 i + 'OOOOO 002 •67381 91 + ■00000 002 — "29472 22 i86 Mr. Ebnest TV. Brown, Theory of the Characteristic ek 2 . Values of i. (etf) ; . («'**■),. 4 — 'OOOOO 012 3 — •ooooo 041 — -00001 274 2 — -00003 797 — •00130 683 j r — •00279 137 —•13063 320 1 o — '01298 640 — •01298 640 1 — i + •29872 990 + ■01621 815 — 2 + -00369 243 + •00008 801 -3 + •00001 816 + "ooooo 059 -4 + •00000 012 Sum + •28662 446 1 -•12863 2 S4 e k ,= + -05385 595 81. Characteristic e'k 2 . Values of K eV + q eV . i. 2i + m + 2g. 28— m- -2g. 5 + 'OOOOO 015 4 — •ooooo 001 + 'OOOOO 870 3 — -ooooo °5* + '00043 519 — 'OOOOO 001 2 — 'OC00I 838 + •01232 953 — 'OOOOO 099 1 + '00038 7658 — 'OOOOO 0271 — '24500 7602 — '00020 8242 + •09673 8490 — •00005 9900 + '09029 2445 -•00529 7319 — 1 -•29235 4798 -•03343 2 7S3 — '00832 9223 + 'OOOOO 0725 — 2 + '00803 763 — •00148 971 — •00018 592 + 'OOOOO 004 -3~ + •00017 185 — -ooooo 704 — 'OOOOO 278 —4 + 'OOOOO 255 — -ooooo 005 — 'OOOOO 004 ~5 Sum j + "OOOOO 003 - -•18703 549 -•03498 972 -•15045 955 -•00550 580 Motion of the Moon. 187 Characteristic e f k 2 . Values of K rV + Q eV . i. 2« + m- -2g. 24- m + 2g. s — 'OOOOO 002 4 — 'OOOOO 210 + 'OOOOO 012 3 — '00011 304 — 'OOOOO 001 + 'OOOOO 679 2 -•00357 5 2 3 — 'OOOOO 105 + -00035 223 1 + •10326 8572 — "00024 0255 + •01107 0715 — 'OOOOO 0040 -•12143 6400 — '00164 8838 — '11412 7077 — 'OOOOO 0725 — 1 + '06401 3724 — '00006 4252 + •10878 3159 + •00529 73 x 9 — 2 + ■00135 509 — 'OOOOO 051 - '00344 466 + •00020 824 -3 + •00002 016 — '00005 398 + 'OOOOO 099 — 4 -s + 'OOOOO 028 — 'OOOOO 067 + 'OOOOO 001 Sum +•04353 104 -•00195 492 + '00258 663 + •00550 S79 i. 2 i + m. 22- -m. 5 4 — 'OOOOO 037 + 'OOOOO 015 — '00003 183 + '00003 380 2 — •00228 182 + 'OOOOO 013 + •00379 855 + 'OOOOO 014 I — '11219 5215 + •00001 8123 + '24359 2093 + '00002 1752 + •01112 0432 + •00210 9609 + •01085 3449 +•00530 5862 — I + •24707 3790 + •03394 4779 — -11769 1816 -•00530 5862 — 2 + '00642 043 + •00014 863 —•00193 573 — '00002 175 -3 + '00010 319 + 'OOOOO 09s — '00002 555 — 'OOOOO 014 -4 + 'OOOOO 141 + 'OOOOO 001 — 'OOOOO 031 -5 + 'OOOOO 002 Sum + •15021 004 + •03622 223 + ■13862 464 i88 Mr. Erxest ^Y. Buowx, Theory of the Characteristic e'k*. Values o£ i. (i*% (v'k'% 016 (**"% <y<t 2 >. 1 4 + 'OOOOO — 'OOOOO 002 3 + '00001 505 — 'OOOOO 389 + 'OOOOO 001 2 — "OOOOO 001 + •00143 Soi — '00047 166 + 'OOOOO 116 I + -ooooo 630 + •10274 226 -•07635 55 + '00008 516 O + •00163 202 + •04278 164 — •08014 891 + '00269 177 -I -•21446 825 — '00081 "5 + •00666 731 + •12718 46 — 2 + •00412 476 — 'OOOOO C73 + '00005 120 — •00186 746 -3 ' + '00001 413 — 'OOOOO 006 + 'OOOOO 039 — 'OOOOO 421 -4 + 'OOOOO 009 — '00000 004 Sum — '20869 096 + •14615 618 — •15026 II + •12809 10 i. 4 (yW)i 0>7*') — 'OOOOO 001 3 + "OOOOO 045 — 'OOOOO 270 2 + '00004 861 — '00025 922 I + -00488 83s — •02176 661 + •48307 75 -•52538 39 — I " + •24918 465 — •08400 495 — 2 + '00065 513 — '00016 048 3 + 'OOOOO 383 — 'OOOOO 081 j —4 + 'OOOOO 003 ! . Sum +•73785 86 -•63157 87 { Motion of the Moon. 189 82. Characteristic e 2 a. Values of jT e =„+ s eV 21", 21 + 20, 2i- -2a. 9 i + •00000 001 — •00001 077 7 ! + •00000 074 — '00082 168 — 'ooooo 035 5 1 _ •00008 103 — 'ooooo 009 — •03612 906 — '00002 740 3 i - •01077 °5° — 'ooooo 799 - •06573 304 4- '00014 077 1 1 — 56564 675 — '00061 101 + •18304 866 — '00221 738 — 1 i + •57575 "8 + '00081 292 — •07199 098 + '00283 9 2 8 J ; _ •02847 660 - '00807 552 — •OO053 429 — '00306 906 [ — 5 ■ _ •00534 300 + '00064 9 I 9 + •00000 433 — '00004 O0 3 ~7 | _ •00006 515 — '00013 766 + '00000 021 — 'ooooo 043 -9 i — •00000 042 — 'ooooo 173 Sum j — •03463 152 — "00737 189 + •00783 338 — '00237 460 9 — •ooooo 031 7 — •00007 051 S — •00628 907 3 — •29381 090 1 — •17424 233 — 1 - •06079 808 -.1 - •04075 79 1 -5 — •00043 4 2 6 -7 — •ooooo 129 -9 + •ooooo 003 Sum — •57640 463 + -f- 00000 004 00000 328 00025 2 5° 00173 425 ■00562 393 00820 455 00126 848 00001 642 ooooo 018 + 00277 397 Royal Astrgsf. Soc, Vol. LIII. d D I go Mr. Ernest V. Beown, Theory of the Characteristic e 2 a. Values of 2J, («H> l'"V)i- («'«)*. 9 — 'ooooo 017 7 + "OOOOO Q02 — '00002 360 — 'OOOOO 100 5 — - ooooo 099 — '00247 899 — -00016 334 3 — '00025 786 - '06279 45 — -01741 144 i — -03029 187 - -07046 35 — -10960 58 — i ' -h "12685 76 — '00347 720 + '21027 9 -3 + ""SSi 33 — '00016 375 — -00049 °°6 -5 •■ — -oooio 571 — 'ooooo 080 — 00007 754 -7 — 'ooooo 946 — 'OOOOO OOI — -ooooo 038 -9 — -ooooo 006 Sum .... + '21170 50 - '13940 25 + '08252 9 $%. Characteristic ee'a. Values of K Re , a + i 2i 2(' + c + m. 28 — 0- -m. 9 + •ooooo 05 + -ooooo 19 7 + •00003 67 — '00008 15 — 'OOOOO 021 5 + '00250 00 + 'OOOOO 004 — '01113 I 4 — "ooooi 929 + •13660 87 + •ooooo 161 — '333 I2 49 - •00150 657 1 + 4-65374 56 - •00038 845 --•15457 19 4- •01267 3 8 4 j — J — •19743 34 + •00791 743 + '93078 13 + •0081 1 527 | ■ -3 — •06923 50 + •09670 148 4- '01512 50 + •00880 652 I -s — '00082 91 - 'OIOI2 338 + -00018 99 + •00009 282 i ~1 + '0000 1 15 — '000 1 1 510 + , 'OOOOO 21 + ■ooooo 091 1 -9 j Sum + ■ooooo 04 — •ooooo 117 + •ooooo 001 + 4'5 2 54° 59 + •09399 246 + '44719 05 + •02816 330 Motion of the Moon. 191 Characteristic eefa. Values of J^+ Q teV 21. 2i'+C- -m. 2i- -e + m. 9 + •00000 06 + •00000 75 7 + •0000 1 84 — • 'OOOOO 001 + •00050 84 + 'OOOOO 001 5 + •00022 6l — •00000 083 + •02742 74 + •00000 061 3 — •01956 28 — •00007 iS5 + •90964 64 - •00004 517 1 — '37493 92 — •00415 929 - •80846 72 + •00545 089 — 1 — •79391 6l + •00386 559 - •07537 94 + •01923 289 -3 + •19032 70 — •01502 54i + •00025 7 1 - •03283 987 -5 + •003 1 1 OS + •00197 73i + •00010 73 — •00044 785 -7 + •00003 94 + "0000 1 999 + "OOOOO 28 - •00000 485 -9 + •00000 04 + •00000 020 + •00000 01 - •00000 005 Sum - •99469 57 - •01339 400 + •0541 1 04 — •00865 339 Values of 28. 0<*)i. («Y«)*- («?'«)(• OV);. 9 + •00000 01 7 + ■00000 05 — 'OOOOO 16 + "ooooo 02 + ■ooooo 96 5 + •00004 4° — '00051 09 + -ooooo 73 + •00099 47 3 + •00454 88 - '05773 38 — '00053 3 2 + •09020 60 1 + •40017 48 - -13184 7 - -04137 23 + •32607 7 — 1 + •01876 3 + '13596 59 - 78645 2 + •01623 91 -3 4- •1 1064 27 + -00107 47 + '01255 04 — '00345 75 ~5 — ■00122 87 + 00000 51 + -00025 44 — "OOOOI ^ -7 -9 •00000 40 + -ooooo 01 + -OOOOO 12 Sum + •53294 I - 'o53°4 8 - '81554 4 + •43005 6 i9- Mr. Eenest W. Brows, Theory of the $4. Characteristic c' 2 o • Values of K^ a + 8 e ««. 2i. 28- 2m. 21- -2m. 9 — •00000 030 + "OOOOO 124 7 — •0000 1 921 + '00003 767 — 'OOOOO 002 5 — •00108 806 — "OOOOO on — •00018 847 — 'OOOOO 286 3 ^•04289 260 — 'OOOO I 784 -•08297 640 — '00026 318 1 — •49418 300 - '00443 201 -•07733 041 —•01694 340 — 1 — •02683 885 + •00651 l62 -•15682 286 + •00130 863 -3 -•01458 690 — •20106 340 —•00657 451 + •00001 217 -5 -f - O0020 815 — •00201 212 — 'OOOIO 781 — '00001 889 -7 + 'OOOOO 792 — •0000 1 913 — 'OOOOO 141 — '00000 025 -9 + ■00000 015 — 'OOOOO Ol8 — 'OOOOO 002 Sum -'57939 270 — •20103 317 -•32396 298 -•01590 780 2i 28. 9 + 'OOOOO l63 7 + '00011 559 s + •00672 559 + 'OOOOO 012 3 + •25523 830 — '00001 306 I + •50942 997 -■00515 710 — I + '*79 r 3 588 —•00450 279 " — 3 + '0435 6 335 + •06839 549 -s + '00066 997 + -00060 357 -7 + 'OOOOO 796 + '00000 560 - -9 + 'QOOOO 007 + 'OOOOO 006 Sum . . . + •99488 831 + •05933 189 Motion of the Moon. 193 Characteristic c /2 a. Values of 2i. (ifa)i. (V"«)i. (irfa)i. 7 — "00000 02 + 'OOOOO II + "OOOOO 15 5 — '00002 75 + - oooo6 42 + '00015 08 3 — '00236 65 + ■00181 74 + •01214 19 1 ~-I0866 00 + •07247 89 + •01287 7 — 1 -■22746 69 + '06296 01 + •24960 2 — 3 -■05932 75 — •00033 14 + •02080 95 — 5 — "OOOIO 54 — "OOOOO 3i + '00004 58 -7 — '00000 °3 + "OOOOO 02 Sum ... "•39795 43 + •13698 72 + ■29562 9 8 z . Characteristic k 2 c Values of E Vli + Q, ■ k? „. 2i. 2S+2g. 24- -2g. 9 — "OOOOO 019 7 — 'OOOOO 039 — "00001 583 — "OOOOO 001 — -00003 127 — '00107 524 — 'OOOOO 139 3 — -00219 2 73 — 'OOOOO 028 — •02025 402 — '00023 440 1 — -10174 53° 2 — '00005 1640 + •07691 4545 -—■00386 8083 — 1 4- '31567 8728 — •01402 7700 -•30052 2 743 -•01234 2006 -3 — •00281 851 -•02319 391 — '00124 397 — '00008 150 -5 — •00082 686 — '001 1 1 281 + '00001 569 — "OOOOO 060 -7 — 'OOOOO 723 — 'OOOOO 681 + 'OOOOO 043 — '00000 001 -9 Sum — •ooooo 007 — 'OOOOO 005 + '20805 637 -.•03839 320 — '24618 133 — •01652 800 194 Mr. Ernest W. Bbown, Theory of the Characteristic k 2 a. Values of l' k% + S? k =„. 2i 21. 9 — 'OOOOO 013 7 — -ooooo 700 S — -00034 902 + '00000 018 3 — •01038 934 + '00002 S77 i + •05314 0982 + '00381 1636 — i + •04282 1667 + •02680 3012 -3 — -03191 SSi + ■02342 170 -5 — •00027 549 + •00014 068 -7 — 'OOOOO MS + '00000 098 -9 + •00000 001 Sum ... + ■05302 471 + •05420 397 Values of 28. (*%)«. (*'*<*)!. (U'a)i. 9 — 'OOOOO 002 7 — 'OOOOO 229 + -ooooo 015 5 + 'OOOOO Oil — •00037 096 + '00002 5°9 3 + •00001 162 -•07251 89 + -00430 805 1 + •00131 775 -•43498 445 + '84123 8x — 1 + •18884 734 -•06346 523 -2-33191 84 -3 + •16090 08 — •00005 398 — "00244 631 -5 — •00041 866 + -ooooo 052 — 'OOOOO 465 -7 — 'OOOOO 086 + 'OOOOO 001 -9 — 'OOOOO 001 Sum ... +•35065 81 -'57139 53 —1-48879 80 Jlotion of the Moon. 195 86. Characteristic ear. Values of X ea * + *-&a- 1 CO Qa" i. 21 + C. 21 — c. 5 + 'OOOOO 05 4 + "OOOOO 24 + '00002 45 — 'OOOOO 02 + '00012 36 — "OOOOO 01 + •00117 89 — 'OOOOO 22 2 + "00520 57 — "OOOOI 19 + •04732 91 — •00023 56 1 + •16389 96 — '00I2I 33 + •07138 99 + '00490 20 — '13821 47 + •00035 22 — •O4822 18 + •00736 36 — 1 + •02858 °3 + •01208 22 + '02221 98 + ■00169 14 — 2 + •00750 77 + '°3393 88 + 'OO062 41 — •00922 16 -3 + •00012 52 — •00184 24 + '00001 36 — '00008 67 —4 + 'OOOOO 23 — 'OOOOI 63 + "OOOOO 02 — 'OOOOO 08 -5 + "OOOOO 01 — 'OOOOO 01 Sum... +•06723 22 + •04328 4i + •09455 88 + '00440 99 ■ ■ Values of i. (« 2 );. (*'«% 4 3 2 1 — 1 — 2 -3 -4 + •00000 22 + •00015 7 1 + •00899 53 — ■01666 28 — •22632 4 + •00943 38 — •oooii 98 — - ooooo 04 + '00000 04 + •00003 JI + •00229 1I + •10151 I — "01666 28 + •00183 95 -•00054 38 — 'OOOOO 2 I Sum — •22451 9 + -08846 4 C^= — -Q22I2 6. 196 Mr. Ernest W. Brown, Theory of the 87. Characteristic e'a 2 Values of K eW >. + Q eV . i. 2«+ m. 21 — m. I 1 5 + OOOOO 02 4 + •00000 01 + OOOOO 56 3 — •00008 40 — -ooooo 01 + 00013 63 — 'OOOOO 07 2 — •01320 15 — •0000 1 42 - 00223 69 — "00008 17 1 — ] •06476 55 — •00261 78 + 04455 94 — '00852 49 + ■23300 345 "•00545 494 + 20737 15s — •00450 490 —1 + •00948 95 — •01227 27 - 21793 97 + •00700 S6 — 2 — •00075 36 -•08834 34 - 00061 S 2 + •01108 38 -3 + •00003 33 •—•00070 66 + OOOOO 45 + -00005 95 -4 -5 + •00000 08 — 'OOOOO 56 + 'OOOOO 01 + 'OOOOO 04 Sum . , . — •83627 74 — • 1 0941 53 + •03128 59 + •00503 7i Values of i, 4 {va-'H- (i'<Oi. + -ooooo or 3 — •ooooo 22 + •00001 15 2 — •00051 33 + "00063 84 1 -•09673 62 + •00542 42 + •05209 5 + •07246 1 __ 1 —•01035 45 — '02641 18 — 2 —•01043 59 + •00131 14 "3 — '00002 64 + -ooooo 27 -4 — -ooooo 02 Sum ... -•06597 4 +'05343 8 Motion of the Moon. 197 •88. Characteristic a 3 Values of /O+S.- 28. [ 2i. j 7 5 3 1 — 1 ! -» i ° I "5 1 ~~ 7 j — •00001 2 ! — •OOO94 2 — 'OOOOI O j — -01711 5 —-00141 9 + ■00371 9 —-00396 6 ; + •00142 4 —-00726 8 — •00256 8 --01393 l 1 -•00013 -'°i353 6 — ■00008 9 Sum ... — •01562 4 --04021 9 i 21. | («"){. 7 S 3 1 — 1 -3 -5 1 ~ 7 + •00000 1 + ■00001 5 — •00125 9 — ■09620 + •27025 -•00349 9 — •00085 8 — 'OOOOO 2 j Sum ... ~ ~ ' + '16845 8q. Characteristic k 3 . Values Df Ak". i? k ». i. 5 (**)* (W-'),-. 2i + 3 2J + g. c- 4 — 'OOOOO 010 — •ooooo 00014 3 + 'OOOOO 007 — •ooooo 744 + 'OOOOO 00009 — ■ooooo 0164 2 + -ooooo 543 — •00049 643 + 'OOOOO OIII — "O0002 1515 1 + -00036 1540 — •02324 4118 + '00001 4145 — •00277 8178 + •01675 5 6 77 — -01731 6970 + •00168 5259 — 1 — •02777 8633 + •00363 2982 -•06974 485 -•01224 585 — 2 + •00186 956 -— '00042 284 — •00158 665 — 00007 8594 -3 — 'OOOOI 059 --•ooooo 958 — 'OOOOO 620 — ■ooooo 0527 -4 — •00000 048 — 'ooooo 014 — 'OOOOO 0039 — 'ooooo 0004 -5 — -ooooo 001 — 'OOOOO 00002 Sum . . . —■00879 744 — •03786 464 — •06963 822 -•01512 483 g v = — -00806 Royal Astros. Soc, Vol. LIII. 62 55- E E 198 Mr. Ernest W. Brown, Theory of the 90. Characteristic ke 2 . Values of j. A <&• Bkt'- 2i + g + 2C. 2J + g- 2C. 2i + g. s + 'OOOOO 036 + "OOOOO 005 4 + 'OOOOO 012 4 'O0O02 174 + "00000 360 3 + 'OOOOO 910 + - ooio6 7Si + •00022 °53 2 + '00056 581 + •03177 374 + •01086 145 1 + •02815 3088 —•00388 7463 +•32763 8723 + •86762 4162 + -00460 5675 + •00651 0052 — 1 -•01571 9318 — •02414 0044 + ■00267 1924 — 2 + ■00148 491 — •00079 507 — "00902 610 -3 — '00085 55° — •0000 1 611 — "00030 517 -4 — ■00002 977 — 'OOOOO 025 — "00000~ 621 -5 — 'OOOOO 063 — "OOOOO OIQ Sum ... + •88123 197 + '00863 009 +•33856 875 Values of 2, (*•*)*. <*«*- )<• (/be )»• 5 + 'OOOOO 00048 + -ooooo 00004 4 + •00000 000 10 + 'OOOOO 0487 + -ooooo 00463 3 + •00000 01 131 + '00004 8331 + -ooooo 4718 2 + •00001 1586 + •00427 6263 + '00046 0150 1 + •00112 1329 + -01672 370 + ■03928 1120 + ■09379 8299 — '08141 572 — 1 — '04400 679 — '00307 9003 — •00716 888 — 2 — "00161 311 — •00003 4589 — •00124 5876 -3 -•00013 55°° — 'OOOOO 0350 — "OOOOI 4005 -4 — •00000 1470 — 'OOOOO 00032 — 'OOOOO 0141 — ■00000 0015 — 'OOOOO 00013 Sum ... + •04917 444 - -06348 088 + •03131 713 • es = +-00318 6183, Motion of the Moon. 199 91. Characteristic keef. Values of -4 kee '- i. 2t + g + C - •-m. 2(' + g— C- -m. 2{' + g + e- -m. 2i + g — e + m. 5 + -ooooo 021 + "OOOOO 001 — "OOOOO 003 4 — -ooooo 014 + •00001 399 + •00000 096 — "OOOOO 205 3 — -ooooi 008 + '00076 2 53 + '00006 148 — •000 1 1 203 2 -•OOOS4 981 + •02945 645 + •00318 821 - -00437 856 1 — '02125 360 + •39674 479 + •10786 825 —•06183 438 -•28354 837 — •26071 788 + •33223 139 + •29819 178 — 1 +•35289 056 + •03417 336 -•03919 252 -•05967 106 — 2 - -01531 540 + •00126 893 + •00654 802 -•00435 546 -3 — •0OIO2 615 + '00002 634 + •00025 093 — 'OOOII 193 -4 — "00002 518 + "OOOOO 041 + "OOOOO 524 — "OOOOO 203 S — 'OOOOO 045 + "OOOOO 009 — "OOOOO 003 Sum + •03116 138 + •20172 9 J 3 + •41096 206 + •16772 422 Values of i. (J«Oi. 0'„ )«. (W)i- (Mv) t . 5 + 'OOOOO 00023 + "OOOOO OOOOI — "OOOOO 00003 4 — "OOOOO 00014 + 'OOOOO 02390 + "OOOOO 00099 — "OOOOO 00336 3 — "OOOOO oiS79 + '00002 3817 + "OOOOO 09940 — "OOOOO 33°° 2 — "OOOOI 5276 + •00220 1482 + "00009 2083 — ■00029 9393 1 — "00132 549* + •15623 236 + •00707 5760 — •02027 447 -■07509 456 + ■22035 892 + •10625 739 -•25455 973 — 1 -■31367 703 + "01194 449 + •03225 857 -•02563 651 — 2 — "01 000 189 + "00009 4212 + ■00276 085 - -00033 4698 -3 — "00009 1842 + "OOOOO 0822 + -00OO2 0390 — -ooooo 3552 -4 — "OOOOO 0874 + "OOOOO 00070 + 'OOOOO 0174 — -ooooo 00351 -5 — "OOOOO 00083 + "OOOOO 0002 — -ooooo 00003 Sum — "40020 7i3 +•39085 635 + ■14846 622 -•301 1 1 172 200 Mr. Ernest W. Browx, Theory of the 92. Characteristic ke' 2 . Values of 1. A ke' - -- Bu.. 2i + g + 2m. 2( + g— 2m. 2i + g. 5 + 'OOOOO 002 4 + "OOOOO 003 + '00000 220 — 'OOOOO 061 3 + 'OOOOO 163 + '00012 938 — "00003 696 2 + '00005 277 + ■00598 953 — •00170 201 1 — •00018 °37 + •15607 842 -•04382 8741 + "00147 449 — '00328 213 +•01075 1875 — 1 + '15457 297 -■00057 354 -•04205 6780 —2 — '00623 77i -•00027 3i3 + •00314 3!4 3 — '00023 118 — •00000 738 + 00009 501 -4 — 'OOOOO 469 , — - ooooo 014 + 'OOOOO 183 -5 — 'OOOOO 009 + 'OOOOO 002 Sum . . . + •14944 78s + •15806 323 -■07363 3 2 3 Values of «'. (for);- (*y% (l- V r, );• 5 + 'OOOOO 00002 — 'OOOOO OOOOI 4 + "OOOOO 00004 + 'OOOOO 00293 — "OOOOO 00079 i 3 + "OOOOO 0032 + 'OOOOO 2898 — "OOOOO 0786 2 + "OOOOO 1954 + •00027 I35 2 — '00007 1657 1 — "00002 8799 + •02119 6537 —•00525 9644 — '00455 928 + •00832 47 — 1 —•25292 577 + •02434 3°4 + •12438 368 — 2 — •00147 2986 + 'OOOOO 4232 + -00064 3737 -3 — 'OOOOI 3*5° — 'OOOOO oi73 + 'OOOOO 5169 -4 — 'ooooo 0120 — "OOOOO 0002 + 'OOOOO 0046 ~5 — '00000 00013 26 + 'OOOOO 00003 Sum . . . -•25899 812 + •05414 + •11970 054 g«,*= +-00564 6 :>■>;>• Motion of the Moon. 20C Characteristic kea. Values of 21. A ea- (/reaV. (/t€'o)j. 2( + g- -c. 2( + g- -c. 9 + 'OOOOO OOI + 'OOOOO 015 + "OOOOO OOOOI + '00000 I 00015 I 7 + 'OOOOO I0 S — 'OOOOO 684 + 'OOOOO 001 1 7 — 'OOOOO 0196 5 + 'OOOOi 277 — '00141 686 + "OOOOO 0096 — '00006 6448 3 -•00397 241 —•10868 994 — 'OOOI7 8806 -■01376 9493 ; i -■40314 29S 4- ; ooo8o 657 -•04580 8867 — '01 190 80 — i + •00752 301 — •01874 8Si + •04850 79 + ■09203 77 ~3 -•03088 873 + •01689 021 + -06482 90 + "00233 1866 j 5 + '00382 908 + •00023 291 + '00067 3777 + "OOOOI 1563 ! _ 7 + '00006 094 + 'OOOOO 141 + 'OOOOO 3486 + 'OOOOO 0041 -9 Sum + 'OOOOO 055 — 'OOOOO 002 + 'OOOOO 0017 — "OOOOO 00003 -•42657 671 — -11093 092 + '06802 66 + •06863 70 94. Characteristic ke'a. Values of 2i. A ke'a- (l-na),. (/.V«) 2 i •■*- £t *+ m. 2S + g- m. 9 + "OOOOO 007 + 'OOOOO ° T 5 + '00000 00008 + •00000 00015 7 + "OOOOO 686 + 'OOOOO 797 + '00000 01232 + •00000 01283 ; S + "00057 193 + "000 28 744 + •00001 82526 + 'OOOOO 8209 I 3 + '039 6 3 137 + 'OO232 832 + '00285 o:t 3 2 + '000 1 3 6733 i ) + rSoi62 I I I — "00466 404 + •51292 301 — •00839 428 j — 1 - -05655 671 + I '79OI3 231 + -04334 669 — 1 ■51928 973 j -3 + '03667 959 - -03946 234 + ■01706 372 — •02090 548 ; — 5 + '0005 1 284 — "00099 194 + •00005 4 2 99 — •00009 7841 7 — 'OOOOO °93 — "OOOOI 460 + '00000 0060 — •00000 0584 \ -9 Sum — 'OCOOO 014 — "OOOOO 016 — '00000 000 1 3 — 1 ■00000 00037 j + i'S2246 599 + 1-74762 311 + •57625 629 •54854 285 202 Air. Ernest AV. Huowx, Theory of the Motion of the Moon. Values of II- . 2 i o — i — .: .■> —4 + "00004 5 7 + ■00228 18 + ■07204 28 + ' 0;! 497 59 t'o-';^ 3.i — , c;c-7^ </) - '00003 63 Sum + ■'-5*. =5 .55 4 1 — i — 2 -3 — 4 Suia lPu-' ifu)il CM ;/■', /'"., U.S.A.: 1899 January 21. ■ 'a 1 . + •00000 0009 + •00009 6^3 + -ooS62 452 ; — '=7777 o - -o-.'i 5?5 ■- ;ocoo 21 j — '00000 002 — O&92S 5 Errata in Part I. Page 45, line 1 1, for " a' " -read " a'." „ 63, last Vmc, for " ;] m 3 " read " ;; nr;'-' 3 ." „ 69, line 9, for " ; " read " , ". „ 75, line 4 from bottom, for " (. . .) " read " (. ,.) 9 ." „ 107, line 10, for " — '00585 014 " read " — -00585 0139." „ „ „ 11, for "— -01024 957 " read " — -01024 9560." » >. » 1 2, /or " +-02515 958" read" +-02515 9581." , 1 ;,./''/■ " + ooS.jS cip "/■•■'"'- —-c.-o'jS =914." ,, 116, Appendix. See corrected values gi-en above. Theory of the Motion of the Moon; containing a New Calculation of the Expressions for the Coordinates of the Moon in Terms of the Time. By Ernest W. Brown, Sc.D., F.R.S. [Received 1900 May 25 ; read 1900 June 8.] PART III. CHAPTER VI. In the first two parts of this Memoir, published under the same title in the Memoirs of tlie Royal Astronomical Society in 1897, 1899, the general theory and the numerical results, up to and inclusive of the third-order terms, have been given. This part contains the numerical results for the terms of the fourth order. The methods adopted are in general the same as those used for the third-order terms. Instead of finding the values of uju directly and then deducing those of u k . as in the third-order terms, I have found the values of u k directly. This change was found advisable when preparations were being made for the calculation of the fifth- order terms. It was seen that the n on -homogeneous equations (17), (18) of Chap. I. would involve much more calculation than the homogeneous equations (6), (7), (8) of the same chapter, and the latter require the results for u k and not those for uju a . In the former case u/r 3 has to be expanded to the fifth order, an enormous piece of work ; in the latter case we only require the calculation of such expressions as u 2 , uBs, &.C.. to the fifth order, and this has been so arranged as to require much less computation. The only other change from the methods of Chap. Y. is that mentioned in 671. As in the earlier work all the assistance I have had in performing the computa- tions has been rendered by Mr. Ira I. Sterner, A.M., and I take this opportunity of again expressing my obligations to him for the ability and accuracy with which he has conducted the work allotted to him.* I have also done a considerable amount of calculation myself, especially in the later portions of the work. * The expense of making the computations necessary to obtain the results given below has been met by a grant from the Government Grant Fund of the Royal Society. Royal Astrox. Soc, Vol. LIY. b Mr. Ehxest W. Brown, Theory of the The following is the table of contents of Chap. V. :— Section (i). A brief outline of the steps followed in the application of the general method to the terms of the fourth order. Section (ii). The final numerical results for the series a, 4 and for the co- efficients of all terms of the fourth order in u, z. Section (i). Formuke and Method of Procedure. 96. The method employed is in general the same as ^\? a ^.^?t^ products in the third line of equation (i 7 ) and in the second line of (18) of Chap. 1. are formed by putting and choosing the part, of the forth order. Tho products in the fourth and fifth lines of (17) and in the third line of (18) are similarly obtained with The additional parts of (17), (18) in which we put „=»!! *> — •*!! are respectively The meanings of the various symbols have been explained in Chap. I. n i ^ r-i 1 nrp treated in like manner. The parts arising from fl, namely, ^ £ ,-5-37, are treatea in In the parts of equations (17), (18), £-! (Z> 2 + 2 m D) % w„, ' -IPtz* we substitute U.. = U» nnd nroceed as follows : in those parts of the arguments of u, z 2 which contain c, g ^« the values of c, g to the second order (that is, we must retam the parte rf these two quantities which depend on rf, e% k", a«) ; when the operates J) B Z vTbeen performed the portions of the fourth order must be retamed. In all other operations the parts c , g» of c, g will be sufficient. The general procedure in performing the calculates and the methods of testing the results are in other respects the same as those explained in § 66. Motion of the Moon. 3 97. The series finally obtained before proceeding to the solution of the linear differential equations are not, in the case of equation (17), the actual right-hand members, but series %, where AC,- The reasons for this have been stated in § 71 (the symbol Z is there denoted by In consequence of this change the series s„ u 2 , s 3 , e 4 (§§ 68-70) must be replaced by series %, % %, h, where (P=2,3>4)- ' U 2 = : | 3— w 2» *P— ~4~a~ S l> Further, as we shall find u^/sl directly, instead of finding uju, first, we use series U x , U s , U 4> U 4 instead of the series TJ l9 TI 2 , U 3 , IT 4 , where a p (p=i, 2, 3, 4). The values of these eight new series are given below. 2 11... ! f »■ 6,. 4 8j. + -ooooo 023 — -ooooo 003 1 — 'OOOOO 002 + - ooooo 003 3 + •00002 287 — -ooooo 402 — 'OOOOO 210 +•00000 300 2 + •00217 032 — •00048 522 — •00020 726 + -00028 048 1 + •17116 620 -•06341 S l6 -•01843 390 +•02151 153 -■29502 375 + ■87776 191 —•87862 801 -■81835 400 — 1 -•00233 SS3 + •01093 391 —•00581 778 —•00739 195 — 2 — •0000 1 9Si + •0001 1 479 — •00004 631 — •00006 191 -3 — 'OOOOO 017 + •00000 113 — •ooooo 038 — •ooooo 053 -4 + •00000 001 Sum — -12401 934 + •82490 732 -•90313 576 -•80401 335 t. u,. tt r u,. u 4 . 4 — 'ooooo 004 3 — -ooooo 022 'OOOOO 608 + 'OOOOO 021 — •ooooo 032 2 — -00002 94*3 — -00089 987 + -00002 939 — •00004 410 1 — •00428 758 — '16212 885 + '00454 712 — •00648 148 -•73409 342 + 2-18355 94 + 1 -•93146 358 — 1 +•43443 130 + -00036 969 + -00869 575 + •03592 927 — 2 + •00015 197 — -ooooo 196 — -ooooo 049 — •ooooo 147 -3 -4 — -ooooo 036 — '00000 001 — '00000 001 — 'ooooo 003 Sum -•30382 777 + 2-02089 23 + 1-01327 197 — •90206 171 Mr. Erxest W. Brown, Theory of the Section (ii). Values of % u^' 1 /^ ; A, izj&%. 98. The following tables show the characteristics, arguments, and types of coefficients of the terms of the fourth order according to the scheme adopted in Section (iv), Chap. I. The numerical results are given below in the same order, and will be found in the 66 given in the first columns. 99 100 101 102 I03 104 105 106 107 108 109 no III 112 113 114 115 Il6 117 Il8 119 Arguments. Types of Coefficients in i^f-'/aA. ee J e 2 k 2 ee'k 2 e' 2 k 2 k* 3 a eVa ee' 2 a e' 3 a ek 2 a e'k 2 a eV e«'a 2 k 2 a 2 21+40 ; 2^+20 ; 2% 2i±(3.c + m) ; 2-i±(3C-m) ; 2i±(o + m); 2i±(c — m) 2i±(2C + 2m) ; 2i±(2C-2tn); 2i±2o; 2i±2m ; 2% 2i±(c + 3m) ; 2i±(c — 3m) ; 2t±(c + m) ; 2i±(c— m) 2t±4m ; 2t + 2m ; 21 2t±(2C4-2g); 24±(2C — 2g) ; 2-i±2C ; 2*±2g ; 2% 2-i±(c + m + 2g); 2*±(c+m-2g); 2 i+(c— m + 2g) ; 2t±(c-m-2g); 2i±(c + m); 2i±(c— m) 2-i±(2m4-2g); 2i±(2m— 2g) ; 2i±2m ; 2i±2g ; 21 2t±4g; 2-i±2g; 21 2*i±3c ; 2ii±c 2i { ±(2C + m) ; 2i 1 ±(2C—m); 2ii±(c + 2m); 2i 1 ±(c-2m)j 2*1 ±c 2» 1 ±3m ; 2i 1 ±m 2i!±(c + 2g); 2ii±(o— 2g); 2ij±(m + 2g); 2'i 1 ±(m-2g); 2"ij+;m 2i±2C ; 2% 2i+(o + m); 2t±(c — m) 2'i±2g ; 21 ke 3 keV kea' 2 Arguments. 2-i + g±3CJ 2* + g±c 2-i + g±(2c + m) ; 2i + g±(2c— m) ; 2-i + gim 2^ + g±(c + 2m); 2t4-g±(c— 2m) ; 2l + g±C (' 4 ),(*' 4 ); (*V),(«' 3 ); (A' 2 ) .(«*>»), («'¥) ; ( £ V), (*'¥ ; ( eV, ?)> ("""') ; (eh'r,*), («^) ( t V),(«'V s );('V),(*'V); ' (* s w')> (*'V) ; («'rM«V 2 ) ; («'W) (^ 3 ), U'n' 3 ) ; H% (*V) ; toV), *W 2 ) ; (eW 2 ), (e'-?V) (v% W 4 ) ; (-N), (nv n ) ; («V 2 ) ( £ 2 M'), ( t '*kk');(ez'k% (e t 'k>*) ; («'**') (£,&»), («'»/*") ;(«jA' 2 ),(€'fl'* s ); ( £ V£ 2 ), (*'^' 2 ) ; ("j'A' 8 ), (*'»**); (,**»). (^' 2 ) ; (>^' 2 )> (^ 2 ) ; , (, S H') ; (rpkK) ; ( W 'F), (^'A;' 2 ) ; (tin'kkf) {¥), (k'*) ; (kW), (K") j («' 2 ) («'<.), (t«a) ; ( t Va), («' 2 a) ( t V),(e'Vo>; («V«).(''V); («V)> ("V«) ( £l? 2 «), ( £ y 2 a) ; ( £! ,' 2 «), («Va) J (,'a), (r,' 3 a) ; (,^'a), ( W "a) (.Pa), (^ ,2 ») ; (* *'M» (^ 2 «) ; ( f tf«), (t'M'a) (»,Po), (^' 2 «) J (^ ,2 a), (i,'A»a) J (ykk'a), (t/kk'a) (A.*)/ (e'V); (e.'a 2 ) («i,a»), ( £ 'V« 2 ) ; (^'« 2 ), (''"/a*) (Fa 2 ), (A;' 2 a 2 ) ; (M'a 2 ) Types of Coefficients in t^/aA.. (^), (A.' 3 ) ; (&V), (A«' s ) (*«»,), (*«' 2 "')-; .(*'>'). (*''*«»); (kei'tj), (kit't]') (kerf), (k- t ' n n ) ; (*^' 2 ) ; (^'»J 2 ) 5 (W), ^''""j') Motion of the Moon. § ] \ I20 ke' 3 i 121 k*e 12 2 kV 123 ke 2 <x 124 kee'a 125 ke' 2 a 126 k 3 a ; 12/ kea 2 | 128 ke'a 2 Arguments. Types of Coefficients in i^/aA. 2i + g±3m; 2i + g±m 2t + 3g ±c 5 2i + g±c 2-i + 3g±m: 2t + g±m 2i l +g±2c; 2i t + g + g±(c4-m); 2i,+g±(c- 2i, + 3g±2m ; 2t,+g 2Ji + 3gJ 2 h+g 2i + g±C 2i + g±m (&,'), (A,") ; {hf-n'), {knn") (Ft), (kV) ; (Vk' t ), (#AY) (k\), (*V) ; (kVc'r,), (¥kW) (F 2 a), (kt'°-a) ; (k-t'u) (ktrjo), (k.-'t]'n) ; (ke.r{a) t (ke'rjn) (*,»«), (WM; (*w'«) (Fa); (Wa) (La 2 ), (F'a 2 ) The coefficients of t-~ x /aX change their signs when the corresponding arguments change their signs. The following long-period terms have been obtained with the required accuracy by the method of Section (hi), Chap. Y. : — Arguments. Coefficients. + ( — 2 + 2c), (A')-» ("' 3 )x ±(— 2 + 2C — 2m), (t 2 lj' = )_„ (t'V), ±(2C-2g), (rk'%, ( t "L') ±(2 + 2m-2g), ('/*% (>)'*)_, The values of W for these terms, obtained by the method of Section (ii), Chap. V., no-reed, as far as they went, with the values obtained by Section (iii) ; this agreement furnished a valuable test. In the solution, of the linear differential equations those sets of terms with arguments 2i have no small divisors ; the continued approximation method was, therefore, employed for such terms instead of the method of Section (ii), Chap. V.* The continued approximation method was also employed in the solution of the differential equations for the terms with characteristics e ri a, eV, ee'a\ &V, and for all the terms in z. The numerical results now follow. The values of % K for the terms arising in u are given in two parts— those arising from the expansion of mEt 1 /^, denoted by ^, and those arising from - ( (I) 2 + 2 mD)^ + — - 1 j , denoted by l\. Then 3 rt( 'o4 £- = -4^-°-_ 1 xpart, characteristic A, in- 3- u d M, .1** . 4V ' 3 rt *o£' f(Z? s +2mD)2^ + ^U'- 1 ( x ' OS ) where A K is the right-hand member of equation (17), Chap. I. * The method for obtaining the results given in the section referred to has been published in the Camb. Phil. Trans, vol. xviii. pp. 94-106, under the title, " On the Solution of a Pair of Simultaneous Linear Differential Equations, which occur in the Lunar Theory." 6 Mr. Ebnest W. Bbown, Theory of the In the case of z, we have where A x is the right-hand member of equation (18), Chap. I., K, is the part arising from the expansion of «/r», and L k is the part arising from the expansion ot , 3ft The numbers are the coefficients corresponding to the power of £ (that is, the argument) which is placed at the head of each column cr pair of columns. The suffix of i x is omitted in the tables. Further details concerning the results will be found in Chaps. I.-1V. Motion of the Moon 99. Characteristic e 4 . Values of M + i. For arguments 2^ + 40, 21, 8=0. i. 21 + 40. 2J - 4c i i 2i„ 1 5 + -0O003 95 + "OOOOO 28 4 + - ooooo 02 + -OOI03 *3 + -00013 62 3 + •00001 08 + •00204 86 + •00587 17 2 + •00059 44 + •00427 24 + •14884 07 1 + •02594 23 -•03457 94 — •06923 47 +•67584 28 + •05789 84 — '06099 65 — 1 —•22643 43 + •00198 63 — •00928 89 — 2 + •00669 89 + '00004 43 +•01353 65 -3 + '00045 68 + '00000 07 + '00045 9 1 -4 + •00010 24 + •00000 99 -s + •00000 34 + 'OOOOO 01 | Sum + ■48321 77 ! +-03274 21 +"02933 ^9 2i t 2C. 2t — 20. 5 + 'OOOOO CI + •00001 79 4 + 'OOOOO £6 + •00077 93 3 + 'C0046 28 + •01582 81 + •00000 10 ! 2 + •02001 44 — 'OOOOO 02 + •00431 95 + •00015 18 1 +•51237 13 — •00002 62 + •00596 01 + •00018 42 — -24760 9i —•00371 37 — •03008 47 — •00041 35 — 1 + •00727 5i + '00098 30 + ■04523 32 — 'OOOOO 16 — 2 + '00121 42 + '00001 06 + •00154 64 — ■? + •00187 28 + •00003 45 -4 + •00006 20 + 'OOOOO 04 -5 + 'OOOOO 13 15 j Sum + •29567 35 —•00275 + •04763 47 — •00007 81 Mr. Ebkest W. Brown, Theory of the Characteristic e * # Values of i. (* 4 )i' (*' 4 )/- («'«% ("")<• («V«)i. 4 +■■00003 9 | + '00001 9 + 'OOOOO 24 3 2 + '00020 -> + 'OOOOO 8 + ■00069 4 + ■00013 21 4--OOOOI 1 + -OOOI4 3 + ■00043 8 + '00034 4 + -00489 88 I + •00054 4 — •O0O77 4 + •01597 r> — •00172 8 — -00639 9 O 4- '02003 5 + -OOI45 3 — '02093 6 + ■00573 7 -•01524 5 — I —•01647 + '00002 5 + •00474 6 + •00118 3 + '00222 8 — 2 4- '00217 + •00039 2 + •00001 9 + "00036 45 —3 4- '00004 + •00005 1 + 'OOOOO 57 -4 4- 'ooooo 3 + 'OOOOO 2 + 'OOOOO 01 Sum + ■00633 3 + ■00108 9 + •00067 3 + -00626 8 — -01401 2 100. Characteristic eY. Values of M+2. 1 i. I 2s + 3e + m, 2i — 3e — m. 1 5 4 3 I 1, 2 1 ' + •00007 26 ! •ooooo 05 + ■00276 44 — •00C02 Si + •05506 64 — -ooooo 21 - •00128 ■04820 •91127 53 09 06 — 'OOOOO — •00023 16 27 + •05714 — •12700 — •10600 34 10 69 — •ooooo + -00002 — -00020 99 49 08 — 1 — 2 -3 + + '44433 •02661 •00794 98 37 42 — •00003 + ■00007 — -00006 60 60 92 — •00429 — •000 10 — •ooooo 96 63 22 + -0002 3 + 'OOOOO + 'OOOOO 40 OI -4 + •00027 94 — -QOOOI 50 -5 + •ocooo 63 — 'OOCOO 02 . Sum | - 1-37027 86 — '00027 87 1 —-12236 92 + '00004 77 Motion of the Moon. Characteristic eY Values of £ + £. i. 2i+3c-m. 2i'-3c + m. 5 4 — -ooooi 05 + "OOOOO 25 i — '00040 3 1 3 : 2 I — I + '00013 + '00624 + -18642 + 1 '09809 - 'i99 J 7 46 15 58 98 10 — 'OOOOO — •00023 + •00019 16 36 86 — •00782 + ■01156 — •06276 + T3395 + •01566 47 09 16 42 24 — 'OOOOO — 'OOOOI — '00007 + ■00117 — •00162 20 09 40 18 22 ; — 2 ; -3 + -00505 — '00101 43 90 — •00002 + 'OOOOO 3° 96 + -00049 + ■00001 22 02 — '00002 — 'OOOOO 77 04 : - 4 — '00003 82 + 'OOOOO 22 + ■00000 02 i -5 — 'OOOOO 02 Sum + 1-09573 01 — •00004 78 | +'09068 02 — ■00056 54 > 21 — c — m. 1. 21 + c + m. 5 „ •00000 03 + •00001 97 4 — •0000: 53 + '00096 49 3 •00077 45 + ■03453 1 5 + 'OOOOO 01 2 '02927 29 — ■00000 08 + •57798 75 + 'OOOOO 04 1 •57"9 50 — •00012 14 + •09193 66 + •00193 3 2 •63687 77 + •00135 94 — '20284 99 + •00026 58 — 1 + •0345 l 56 + •00209 42 -•06986 67 — •00025 82 — 2 + •07900 65 + '00021 27 — -00267 69 + •00014 10 , -> + •00332 03 — '00020 68 — •00006 43 + •00000 24 -4 + •00008 35 — 'OOOOO 34 — 'OOOOO 09 -5 Sum + 'OOOOO 16 — "OOOOO 01 . — •I2I20 82 + •00333 38 + •42998 15 + •00208 47 Royal Astros. Soc, Vol. LIT IO Mr. Ernest W. Brown, Theory of the V Characteristic e 3 c y Values of Jt + &. i. 214-c— m. 2i — c + m. 5 + "OOOOO 17 — ■ooooo 33 4 + -00008 90 — •00015 73 3 + '00426 64 — ■00593 J S — -ooooo 02 2 + -14161 61 — 'OOOOO 22 — -11616 23 — -00003 40 I + 178220 15 — '00030 33 — -03064 08 — •00029 97 O + -20914 40 — •00135 79 + •07494 72 — ■00056 43 — I - -01-55 70 — '00034 43 + •23267 55 + •00165 22 — 2 — -01469 83 — -00003 47 + •01289 11 — •00098 79 -3 — '00055 16 + '00002 95 + '00035 40 — -ooooi 68 -4 — 'OOOOI 36 + 'OOOOO °S + 'OOOOO 77 — -ooooo 02 -s — 'OOOOO 04 + 'OOOOO 02 Sum + 2'iio49 78 — '00201 24 + •16798 05 —•00025 09 Values of i. | 4 3 2 1 ■ —1 —2 -3 ' —4 1 (c'V),. + "OOOOO 2 + •0001 1 + '00445 3 +'05193 I — •02861 3 — •00166 — '00004 7 («'*>»>«■ — •00001 3 — '00040 9 + '00307 6 -•00913 3 + •00576 7 + •00024 4 + 'ooooo 5 — '00002 4 — -oo 1 2 5 5 — •O4042 4 — -06117 4 + -00900 2 + "00036 + -ooooo 4 + -00008 7 + •00321 8 + -00936 8 . --00590 5 — •00396 9 — -00006 1 — -ooooo 1 Sum -•09351 1 + •00273 7 + •02617 6 — -00046 3 1 Motion of the Moon. if Characteristic eV. Values of i. 0V„).. («'Y) 4 . (A'A- (ee' s 7)) 4 - 4 +•00002 + 'OOOOO 1 — -oocoo 2 3 — -ooooi 5 + •00101 2 + ■00008 3 — •00017 2 — •00080 + •03124 7 + •00380 -•00593 2 I — -02711 8 + ■00408 + ■09003 4 — •01970 4 O -•08633 — •04425 8 + •01113 1 + -07463 4 — I + ■03812 8 -•00268 8 + •03997 4 + •01114 3 — 2 -3 + ■00390 + ■00004 1 9 — -00003 8 — 'OOOOO 1 — -00060 — 'OOOOO 8 + ■00019 + 'OOOOO 9 4 -4 + 'OOOOO 1 — 'OOOOO 1 Sum — ■07218 4 — •01062 6 ; +'i444i 4 + ■06017 2 1 o 1 . Characteristic ere 1 Values of £ + &. 2,/2 21 — 20- -2111. *. 22 + 2C + 2m s 4 3 + -00004 49 + •00204 02 + •00001 55 + •06459 °5 — ■00000 10 + •00057 27 + •86143 06 — •O00l6 58 1 + •00793 85 + 'OOOOO 40 -•47189 56 — •OOO28 97 -•37149 07 — '00042 61 —•07081 97 — •00 1 00 5 1 89 — 1 — •66500 19 — '00105 94 + -00030 3° — -00034 — 2 + •20280 05 — •00298 76 + -00004 17 — ■00000 73 -3 + •00849 83 -■00133 65 + 'OOOOO TI — 'OOOOO 01 —4 + •00021 90 — •00002 05 -s + 'OOOOO 49 — 'OOOOO °3 Sum — •81644 3 2 — -00582 64 + •38573 67 — •00181 79 12 Mr. Ernest W. Brown, Theory of the Characteristic eV 2 . Values of M + %. 1 * 22 + 20 — 2111. 22- i -20 + 2m. 1 5 — 'ooooo 05 4 + '00001 24 — 'ooooi 73 3 4- '00072 90 — '00278 01 + -OOO00 02 2 + '02854 II — -ooooo 02 —•18735 97 — •ooooi 29 I + '599°4 28 — -00002 69 + ■14699 65 — '00046 45 O + 1-26100 51 — •00126 II + •25718 37 + •00742 25 — I + "^oS 3° + -00089 5° + ■06791 35 — - oi2i8 97 — 2 - '03928 35 + •00015 55 + •00269 °3 — '00031 19 3 - '°°°52 55 — -ooooi 86 + •00006 59 — •ooooo 46 —4 — "ooooo 42 — -ooooo 03 + 'boooo 07 — •ooooo 01 -5 — "ooooo or Sum + 1-98857 01 — •00025 66 + •28469 30 — •00556 10 ;' i 2J + 2C. 2t- -2C. 5 — 'OOOOI 20 4 — •ooooo 48 -•OOO58 51 — •ooooo 01 3 — •00024 76 — •O1832 28 — 'ooooi 42 2 -•01039 93 + •00000 27 -•23064 49 — •00208 94 1 — •25413 96 + •00031 50 -•O2504 78 — -00311 93 -•54218 73 + •04731 79 — •O9083 51 + •00876 11 : — I . — •02036 16 — '01141 91 — "02701 79 + '°°555 46 i —2 -'04993 83 — •00067 12 — •00096 10 + •00010 91 ; -3 1 — •00225 J 6 + •00038 18 — '00002 02 + •00002 15 ! ~ 4 — •00006 08 + •00000 59 — 'ooooo 05 + •00000 01 -5 — "ooooo 07 + •00000 01 Sum -•87959 16 + '03593 3i -"39344 73 + •00920 34 Motion of the Moon. 13 Characteristic eV Values of & + t 2i-2m. i. 21 + 2m. 5 + -ooooo 79 + -00036 03 4 3 2 4- -ooooo 27 + "00002 &Z + '01554 27 — -ooooo 01 06 89 — -OI244 90 + •00000 16 + -41891 21 — -ooooi 1 -I-D3332 56 — ■00017 67 + 3-40300 52 — •00103 — -11065 41 + •00096 19 - -10257 34 + •00056 05 — 1 + -75°79 48 + ■00658 16 — -20699 71 — -00117 71 26 — 2 + -05144 80 - -00866 61 — -00256 42 --•00016 — 3 + -OOI57 54 — -00016 17 — -ooooi 78 — -ooooo 31 —4 + -00003 37 — -ooooo 22 + -ooooo 03 -5 + -ooooo 08 Sum - -35254 48 — •00146 16 j +3-52567 60 — •00183 19 i. si. 5 — 'OOOOO 19 4 — -oooii 21 3 — -00497 96 — -ooooo 01 2 - •I4 I S5 47 — -ooooo 92 I _, -23587 3 2 — -00076 02 — -3976° 68 — -00044 40 ! 1 —I — -24832 43 + •00439 86 i — 2 — -01642 02 + •00294 3° 1 _ 3 — -00048 60 + •00005 04 ! -4 — -ooooi 16 + -ooooo 07 -5 — -ooooo 03 _ ! Sum i j —2-04537 07 + ■00617 92 14 Mr. Ernest W. Brown, Theory of the Characteristic eV 2 . ^ /alues of «'. ; OV)i. 1 1; (<V 2 ) 4 . | OV).. 4 + '00005 3 1 — 'OOOOO I 3 -+ 'OOOOO 2 + •00275 9 + •00001 2 — "ooo 1 6 I 2 + "OOOOI + •08685 8 + •00062 3 — '01601 I + '00011 + •07421 6 + •02043 7 — '14196 O — •02678 9 — '00684 1 + •10603 8 + •03200 2 — I -•28583 I — 'OOOOO 3 + •21767 + ■00135 S — 2 + •02348 4 + "OOOOO 1 — •00396 8 + "00003 4 -3 + '00009 9 — 'OOOOI + 'OOOOO 1 -4 + 'OOOOO 4 Sum -•28891 1 + •15704 3 + •34080 —•12474 i. 0V)«- I (6'W) . («V)<. («y% («W)i- 4 — "OOOOI 5 + "OOOOO 7 — "OOOOO 20 3 — 'OOOOO 4 — •00072 8 — 'OOOOO 3 + •00037 8 — 'OOOII 78 2 — '0002 2 4 — '02068 2 — •00073 6 + ■01603 8 -•00514 67 i — ■00815 4 + ■05116 -•07832 4 + •30827 9 — '10410 4 -•03958 2 — "00131 - -18186 + '16309 —■09951 9 — 1 —•05663 —•00042 6 + -I °559 3 -•02533 — '02118 2 — 2 — '00702 8 — "OOOO I 1 + •00083 1 — '00004 7 — '00O2I 77 —3 | — '00002 2 + '00O02 1 — 'OOOOO 58 — 4 — "OOOOO 2 — 'OOOOO 01 1 Sum -•11165 + •02799 -•I5448 + '46242 — ■23029 1 s ! Motion of the Moon. 15 102. Characteristic ee n . Values of .ft +?. i. 21 + C + 3m 2t-e- ~3 m - s 1 + •00001 11 4 1 + ■00057 24 3 — 'OOOOO 23 + •02308 42 — 'OOOOO 02 2 + •00001 06 + •52094 60 — '00002 74 1 + •00353 66 + •00002 17 + •26017 23 — •00414 10 — •04803 63 — -00372 49 — •01662 47 + -00308 04 — 1 + ■13629 36 + •13899 01 + '00047 49 - -00034 45 ! — 2 + '09169 96 — '04090 63 + 'OOOOO 48 + -ooooo 03 -3 + '00306 65 — •00057 78 — 'OOOOO 01 -4 + '00006 57 — •00000 7i -5 + 'OOOOO 11 — 'OOOOO 01 Sum + •18663 5i + '09379 56 + •78864 09 — •00143 24 i. 21 + c — 3m. 21- -e + 3111. 5 4 + "00004 19 + 'OOOOO 09 \ 3 + '00205 95 -j- 'OOOI2 66 2 + "07211 99 — 'OOOOO 12 + •00832 00 + 'OOOOO 13 ! I + 1-18317 59 — •00015 04 + •10347 72 — '00023 73 O + '27486 46 -■00493 83 + ■11154 53 + •02718 3i — I + '03165 59 — ■00125 28 -I -20318 44 -■06705 07 — 2 + '00139 35 -•00057 45 + ■00898 11 — '00176 45 : —3 + -0OOO2 18 — - coooo 18 -i- - O0O22 34 — -CO0O2 60 — 4 + 'OOOOO 02 + XOOOO 45 — 'OOOOO 03 Sum + 1-56533 32 — '00691 90 + •43586 34 — '04189 44 i6 Air. Ernest W. Brown, Theory of the Characteristic ee n . Values of M + 2- i. 2? + c + m. 2b — ( — m. 5 — -ooooo 44 4 + 'OOOOO 13 — -00024 16 | 3 + 'OOOIO 29 — -00971 52 — -ooooo 34 i 2 4- -00302 38 — -ooooo 20 -•21574 67 — -00041 10 | I 4- -01409 57 — -00027 39 -•37164 88 — -02742 72 O -•I549S 99 — •01772 85 — -04966 63 + '02712 44 ; — I -•I594S 22 -•03853 74 4- -OOI45 78 — -00097 °s 1 — 2 -■03562 96 + •01719 29 4- -00029 53 — '00008 42 | -3 — 'OOI22 46 -f '00024 26 4- -ooooo 94 — 'ooooo 14 ! -4 — -00002 66 4- '-ooooo 3° -5 — -ooooo 04 4- 'ooooo 01 Sum — •33406 96 -•039 TO 32 -•64526 °5 — •00177 33 i. 2i + C- -m. 2I- -c + m. s 4 — -ooooi 78 4- "ooooi 34 3 — -00094 85 4- 'ooooo 01 — 'OOCOI 01 4-'oooco °5 2 -•03389 63 4-'ooooi 58 -•03797 42 4- -00006 02 1 -•56836 53 4-'ooi96 08 — -14814 17 4- -003 1 8 13 — '19411 40 4- -01863 26 -•06658 97 4- -02765 6 s — 1 — •05160 44 4- -0049 1 80 — •09074 64 + 'o3547 34 — 2 — •00689 49 4- -00272 61 — •00401 73 4- -00083 57 -3 — -00005 02 — -ooooo 95 — -00009 86 + •00001 19 -4 -5 4 -'ooooo or. — -ooooo 03 — -ooooo 13 + -ooooo 01 Sum -•85589 13 4- '02824 36 -'3475 6 59 + •06721 96 | Motion of the Moon. *y Characteristic ae n . Values of i. (^ 3 )»- OV 3 );. («)' S )i- OVJi. 4 + •00001 4 + -ooooo I 3 + ■00078 6 + -00004 2 + -ooooo 4 2 — -ooooo I + •03522 I ] +-002I2 8 + •00037 4 I — -00007 8 + •78390 J +-066I3 8 + •00357 O — -04064 + -IIIII 1 +'I73 8 5 -•28327 — I — 1-52648 — -OOOOI 2 j +-03836 + -00097 5 — 2 — -00492 4 —■00003 4 + •00014 7 -3 + -00004 7 + -ooooo 3 —4 + -ooooo 1 Sum — 1-57208 + •93102 +-28049 — -27820 i. («jV)j. (eW 2 ),:. (fliA- (eW)i- 4 — - ooooo 6 3 + -ooooo I — •00031 1 — •0000 1 9 — 'ooooo 5 2 + -00006 4 -•01336 8 —•00094 1 — -00116 8 1 —•00045 I — •28623 — ■02764 6 + •12094 -■19733 + •54293 + •23524 -■74298 — 1 + "50408 — •00002 3 — •41624 + -00049 2 ! — 2 + •00184 5 + 'ooooo 3 — '00008 8 — -00005 7 j -3 — 'OOOOI 6 — -ooooo 1 — -ooooo 1 1 -4 — 'OOOOO 1 Sum + •30819 + ■24300 — •20970 — ■62278 Royal Asteon. Soc, Vol. LIV. D Mr. Ernest W. Bbowx, Theory of the 103. Characteristic e /1 Values of M + %. i. 21 + 4m. 21— 4m. s + 'OOOOO 13 4 + OOOO5 52 3 + •002 57 71 2 4- "0000 1 02 — 'OOOOO 01 + '07920 62 — 'OOOOO 22 1 4-'ooi32 73 + •00014 6S + •13198 73 —'00036 II + •03140 86 —•00886 25 + •02357 85 —•02153 32 — 1 4-'02695 70 -•39736 75 + •00022 85 — •00052 41 — 2 + ■01319 27 -•00549 20 + 'OOOOO 28 + 'OOOOO 07 -3 + -00035 92 — -00006 61 -4 + -ooooo 75 — 'OOOOO 08 -s + 'OOOOO 01 Sum + •07326 26 — •41 164 25 + •23763 69 — •02241 99 i. 2.1 + 2m. -28 — 2m. 5 4 — •00003 04 3 — 'OOOOO 81 — •00143 50 2 — '00003 41 + 'OOOOO °3' -•04376 96 — 'OOOOO 10 1 — '00080 80 + 'OOOOO 32 — •15494 59 — '00034 21 ; — '02145 3° — '00619 14 — •01687 08 + '00132 00 — 1 ! — '03026 82 + •24551 88 — 'O0020 08 + ■00031 30 — 2 ! — '00693 33 + •00294 84 — 'OOOOO 49 — 'OOOOO 01 -3 1 — '00019 14 + '00003 64 — 'OOOOO II + 'OOOOO 02 — 4 -5 ■ | — 'OOOOO 35 + 'OOOOO 04 \ j Sum -•05969 96 + •24231 61 -•21725 85 + -00129 00 Motion of the Moon. 19 Characteristic e 14 . Values of £ + &. i. 2i. 5 4 + -00000 53 3 4--ooo2i 42 2 + •00536 69 +-ooooo 10 1 + •01053 21 — -0O004 58 — •01872 34 —-00267 3 2 — 1 + -00I69 56 — -0II29 18 — 2 + •00078 57 — -00037 20 -3 + •00002 50 — -ooooo 57 -4 + •00000 04 — '00000 01 -5 Sum — '00009 82 —-01438 76 Values of (i 4 )f. 3 2 1 o — 1 — 2 -3 Sum + "00001 o — •02698 — •61854 2 + •00019 2 + •00000 6 -•64531 (n'% + -00007 5 + ■00398 6 + •14687 7 + 04747 — "00019 2 + 'OOOOO I + ■19822 (»y)<. + 'OOOOO I + •00018 I + •05634 + •26249 4 — '00006 7 — -ooooo 3 + •31895 (Wk (nW"-)i. 50 — -00003 8 + 'OOOOO — -00198 9 + •00019 44 j — •06580 + •00245 9 i —•08045 -•00536 i — •00019 8 -•00885 8 ! + •00000 37 i —•14848 + •00000 03 -•01155 6 20 Mr. Eknest W. Brown, Theory of the 104. Characteristic e 2 k 2 . Values of M + %, For arguments 2i±2c±2g, 21, g=o. i. | 21 + 2C + 2g. 2J — 2C-2g. 2J + 2C — 2g- 2i — 2C+2g, 5 + 'OOOOO 50 — 'OOOOO 07 4 — 'OOOOO 01 + '00O2I 12 + - O0000 II + 'OOOOO II 3 + •00000 02 + '00430 10 + •00002 24 + 'OOOOO 41 2 1 — 1 + '00007 + '00500 + •22308 -•69815 85 53 59 36 + "02174 — '18229 + I'I7697 + "03394 09 57 36 87 -•OOI75 — ■27258 + "53053 -•I5877 OO 99 65 59 —•00591 -•86503 + "55 OI 5 -•05478 44 82 59 89 — 2 -3 -4 + '03225 + '00096 + -00006 26 80 01 + '00062 + "OOOOO 72 89 + •04235 + '00128 + '00002 33 22 42 + •00037 + '00002 + 'OOOOO 20 35 10 -5 + "OOOOO 17 + 'OOOOO °3 Sum —•43670 14 + 1-05552 08 + •14110 42 -•37518 46 *. 28 + 2C. \ 2S- -2C. 5 — "OOOOO 14 4 — 'OOOOO 04 — -OOOIO 32 + 'OOOOO 01 3 — -00002 44 -•00525 50 + '00002 08 2 — -00123 16 — 'OOOOO 45 —•16771 28 + '00304 27 1 — -02814 68 —•00052 60 + '04896 69 + 'OO369 3 1 + 1-68846 77 - "07456 96 + •68013 38 — '00829 12 — 1 + "04541 74 + '01971 2 5 + "07005 51 — -OOOO3 14 — 2 — '03106 66 + •00021 32 + •00187 82 — "OOOOO °5 -3 + '00182 12 + 'OOOOO °5 + '00003 40 —4 + '00006 20 + 'OOOOO 08 , -5 + "OOOOO r 9 Sum + 1-67530 04 -•05517 39 + •62799 64 — •00156 64 _ Motion of the Moon. 21 Characteristic e 2 k 2 . Values of M + %. i. 2i + 2g. 21- -2"". 24. 5 + '00000 01 + 'OOOOO 04 — 'OOOOO 04 4 + 'OOOOO 09 + •00002 74 — 'OOOOI 39 3 + 'OOOOI 91 + •00068 73 — '00076 69 2 + -00099 15 -■03325 97 + 'OOOOO 13 — -03247 94 I + '01302 76 — 'OOOOO 01 + -28101 50 + '00076 36 - -57621 43 o -4-51169 89 — -00009 89 -•67308 93 — '01560 86 -1-44031 63 — I + -35823 7 1 + •00168 86 + •44117 02 — '00003 52 - '05952 65 — 2 — -00820 20 + 'OOOOO 54 + •01307 25 — 'OOOOO 01 + -02225 23 -3 + '00035 II + •00024 35 + -00067 3i —4 + 'OOOOI 18 + 'OOOOO 49 + -ooooi 20 -5 + 'OOOOO 02 + 'OOOOO 01 + - ooooo 01 Sum — 4-14726 IS + 00159 5° + •02987 2 3 — •01487 90 -2-08638 02 Values of i. («***)«■ U"-k'% (.'&'»),. («'«**)«• 4 3 2 1 — 1 — 2 -3 —4 + •00002 4 + •00085 7 — '04605 1 + ■01017 3 + '00009 1 + '00000 4 + •00001 + '00042 2 + •00109 5 — •03189 + '09359 9 + •00112 + 'OOOOI I — ■00017 — •02398 8 -■07730 — -0I02I 8 + ■00422 4 + -00004 8 + -ooooo 1 — -ooooo 7 — •00079 ° -•07175 9 + '34043 -•00454 5 + ■00008 9 + 'OOOOO I Sum 1 - '03490 2 + •06436 7 — -10740 + •26342 22 Mr. Ebnest W. Bbow^ t , Theory of the Characteristic e 2 k 2 Values of i. 4 («'**'), {*>H:k'). ("'k% («'*")«■ («'*£')«• — 'OOOOO 4 + 'OOOOO I — 'ooooo 03 3 — 'OOOOO I — •00027 1 + 'OOOOO 7 — 'oooo2 43 2 — •00003 2 — •01602 8 — '00003 2 -■00323 2 — '00151 42 I — •00102 5 -•03306 — '00370 3 — '02056 -■05637 1 O 4--o88j3 1 + •25316 -'35724 2 + •01475 2 - -36003 — I + •05520 + •00622 2 + '16072 + •03914 8 + "°3S25 8 — 2 + •00101 8 + '00006 9 — ■00146 8 + '00045 8 + ■00247 °4 -3 + •00026 3 + 'OOOOO i + '00003 8 + 'OOOOO 5 + •00002 72 -4 + 'OOOOO 2 + 'OOOOO 1 + 'ooooo 03 Sum + •14416 + '21009 —•20169 + •03058 — •38018 4 105. Characteristic ee'k 2 . Values of & + %. i. 2i + e + m + 2g- 2« — e- -m-2g. 5 + 'OOOOO 42 4 — "ooooo °5 + •00021 14 3 — -ooooo 08 + •00877 60 2 — '00006 92 + •16983 13 — '000Q2 30 1 - -00356 07 + 'OOOOO 01 -'34429 60 + '000l6 72 - -08618 70 + '00005 01 -'36413 27 — •OOI79 20 — 1 ~ ri 9953 59 -•00937 10 -■02535 01 + 'OOOOO 33 — 2 + "07 743 85 + '00064 02 — '00060 82 + '00000 01 -3 + -00306 6 9 — '000 1 5 20 — •0000 1 04 -4 + • 00008 06 — 'OOOOO 16 — 'OOOOO 03 -5 + 'OOOOO 20 Sum — 1-20876 61 -•00883 42 -'55557 48 — '00164 44 Motion of the Moon. 23 Characteristic ee'k 2 . Values of §, + Z. i. 2J + e + in- 2g- 21 — < ; — m + 2g. 5 + -ooooo 08 4 — 'ooooo 21 + '00000 08 3 — 'ooo 1 8 74 + •00008 57 2 — '00927 28 — •00000 20 + -00432 54 1 -■30774 52 + •0002 7 64 + •09701 39 — '00002 o7 + ■00314 35 — ■OIO50 06 —•00505 45 + -00286 25 — 1 + •70749 5i + ■01781 40 -•10767 95 + •00265 37 — 2 + ■03875 60 + ■00005 33 —•00459 77 — •00016 85 -3 + •00089 18 + ■00000 02 — •000 10 12 + 'OOOOO 02 -4 + ■00001 5i — 'OOOOO 13 -s + '00000 03 Sum + •43309 43 + -00764 13 — •01600 76 + •00532 72 i. 2J + C- -m + 2g. 2(- -c + m — 2g. 5 — 'OOOOO 04 1 1 4 — 'OOOOO °5 — ■00004 96 3 + 'OOOOO 80 — '00252 47 2 + '00045 8l — •06776 15 — •00002 05 ! 1 + -02115 l6 + '09205 61 — "00026 88 ; + •21254 26 — '0000 2 l6 + •47352 52 + •00138 82 — 1 + •30194 12 + '00179 35 + •13623 37 — *oooo6 5° j — 2 — •02729 45 — 'OOOI4 91 + •00367 59 — '00000 08 — 3 — '00103 08 + 'O0O02 !5 + '00006 67 -4 — '0000 2 3 2 + 'OOOOO 02 + 'OOOOO 03 ~~ 5 — 'OOOOO 01 Sum -^•50775 24 + 'OO I 64 45 + •63522 17 + '00103 3t 2 4 Mr. Ernest "W. Brown, Theory of the Characteristic e/k 2 Values of M + 8. 1. 2i'+c — m- -2g- 21- -c + m + 2g. s + 'OOOOO 03 A + - OOOOI 31 — 'OOOOO 08 3 + "00054 61 — "OOOOO 82 2 + •01389 61 — 'OOOOO 20 — •00027 59 I + -4795 2 76 + •00019 55 + •00611 99 — 'OOOOO 46 O -'°3394 87 — •00262 24 -•17243 46 + -00074 49 — I —•08746 22 -•00288 95 + •18225 78 — •02624 26 — 2 - "0053° 01 — 'OOOOO 88 + ■00774 06 + •00118 16 -3 — •00012 49 + •00027 29 — 'OOOOO 14 -4 -s — 'OOOOO 19 + 'OOOOO 73 Sum + •36714 54 -■00532 72 + •02367 90 —■02432 21 i. 2» + c + ra. 2i — c — m. 1 5 — 'OOOOO 18 4 + 'OOOOO 01 — -00006 09 3 + '00003 75 -•00323 90 + 'OOOOO 50 2 + '00182 63 + 'OOOOO 37 -•10552 73 + '00060 49 1 + -04846 15 + '00044 97 -•59901 89 + -03886 04 + 1-24970 06 + '02492 64 + •37321 43 + -00038 74 — 1 - -30203 78 + •00957 00 — -11431 89 — '00019 28 ~ 2 + -06215 35 + -00498 5i — •00316 98 — 'OOOOI 52 -3 + '0=254 52 + -00004 12 — '00005 58 — 'OOOOO 02 -4 + -00005 27 + 'OOOOO 04 — 'OOOOO 05 -5 Sum + -ooooo 06 + 1-06274 02 1 95 + •03997 65 -■45217 86 + '03964 Motion of the Moon. 25 Characteristic ee'k 2. Values of & + 1. i. 2i + c — m. 28- -c+ m.. 5 + -ooooo 03 — 'OOOOO 03 4 — 'ooooo 41 + •00000 95 3 — -0002I 94 — "OOOOO 02 + •00052 42 — 'ooooo 07 2 — -01004 62 — '00002 58 + •01909 27 — -00008 59 1 — -19834 06 — -00318 89 +•45975 74 — •00548 S7 — 1-20621 23 -•03223 59 -•47842 15 — ■01404 66 — 1 + '16517 35 — -0020I 12 + •20668 08 + •00052 86 — 2 — -02140 03 -•OOO73 39 + •01059 82 + ■00010 47 -3 — -00061 57 — 'OOOOO 60 + •00024 04 + •00000 13 —4 — -ooooi 18 + 'OOOOO 40 -5 1 — -ooooo 03 Sum ' —1-27167 19 — •03820 19 + •21848 54 -■01898 73 Values of i. + -OOOOO 2 + •00079 3 -•14433 + -04641 + -00043 6 + -ooooo 4 («','*"% («,*'% (ey£ 2 )i. 4 3 2 1 — 1 -3 -4 + -ooooo 7 + -00047 5 + •01895 -■08133 — •05729 2 — '00121 6 — •ooooi 5 — '12042 — •ooooo 5 — '00020 I + •01155 -•13724 + •15104 I + ■002X6 2 + '00002 2 — -ooooo I — 'OOOII 6 — •OIO09 I + ■04755 — •18609 — •00070 8 — 'ooooo 4 + •02733 -■14946 Sum — -09669 Royal Asteon. Soc, Vol. LIV. 26 Mr. Ernest "VY. Brows, Theory of the Characteristic ee'k 2 Values of i i- O,'* 2 ),. («V 2 )i. ("I"*"),. (>'#•% \ 4 3 2 I O — I ' — 2 —3 —4 + 'OOOOO 2 + ■00014 ° + '002l6 2 + •03509 — '00641 — '00013 7 — 'OOOOO 1 — 'OOOOO 2 — 'OOOI5 I — '01142 + •03952 + •08429 + •00694 7 + •00009 2 + 'OOOOO 1 + 'OOOOO I + '00000 8 — '00028 7 — •12043 + •14853 — 01646 7 — '00027 5 — 'OOOOO 3 + •00002 + •00146 6 — •07174 + •52873 + •00182 4 + '00001 3 Sum + •03085 + •11928 + •01108 + •46031 i. («?K-')i. (*'i)'M:') ( . (ei;'M% («W)i- 4 — 'OOOOO I 3 + 'OOOOO 2 — '00015 — 'ooooo 6 + 'ooooi 9 2 + "00009 7 — '00960 3 — '00039 1 + '00082 7 1 + '00671 7 - '35943 -•01857 9 — "°833° + '46033 — '82002 -•65366 + 1-19699 — 1 + -59651 — '02285 7 + •46862 + -05037 5 — 2 + "02033 9 — '00017 9 —•00525 9 + '00064 8 —3 + '00017 6 — 'OOOOO 1 — -00003 9 + -ooooo 7 —4 Sum + 'OOOOO 2 — "OOOOO 1 + 1*08422 — 1*21224 — '20932 + 1-16557 Motion of the Moon. 27 106. Characteristic </ 2 k 2 . Values of M + i- ! 2i— 2m- -2g. i. 2J + 2m + 2g. _____ -_-—-- 4 + -00004 18 + •00224 16 3 — 'OOOOO °3 + •07521 17 + 'OOOOO 14 2 + 'OOOOO 75 1 + -00043 °3 — 'OOOOO 08 -•5399 8 50 — •00079 — -00969 35 + •09250 43 + ■00101 26 + •07918 70 97 —•68816 3 1 —■09327 73 — •00013 90 + -00002 11 — 2 + •03528 74 -•00795 57 + -00005 74 + -ooooo 01 + ■00101 97 — -00005 78 + 'OOOOO 12 -4 + •00001 95 — -ooooo 05 __ Sum -•5S889 47 — ■10027 95 -•38338 33 — ■01047 06 . 11- - 2m + 2g. i. 2! + 2m- "2g- , — — — -ooooo 02 4 + -ooooo 20 + •0000 1 86 3 2 1 — 1 — 2 + -00006 94 + -00203 + •12148 — -12571 + •21424 + •00727 74 35 86 + -ooooo + •00010 -•00395 01 79 81 + •00116 + -04889 -•11524 3 2 60 84 — 'OOOOO + •00001 04 3° 14 26 — -00030 — '00000 39 43 + ■12137 — '00002 + '00002 41 34 67 —•00037 — ■00042 — -ooooo 3i 17 17 -3 -4 + •00014 40 + 'OOOOO 18 ] + 'OOOOO 11 Sum + •21953 35 -•00415 83 1 +'05620 77 — •00078 39 28 Mr. Ernest W. Brown, Theory of the Characteristic e'W Values of &+%. i. 2! + 2m. 2i — 2m. 4 — '00000 08 — "OOOOO 89 3 — '00002 is — "00047 64 2 — ■00156 93 — •00000 01 -• OI S35 04 I — -11505 II — - ooooo 61 + -44404 18 + 'OOOOO 55 O -•17343 72 — ■00016 14 -•13677 02 + •01266 87 — I + •52334 25 + •"337 32 — 'io334 50 + •00245 57 — 2 + •02473 90 + •00102 26 — •00042 17 + '00002 21 —3 + •00058 62 + 'ooooo 90 + 'OOOOO 55 + "ooooo 02 —4 + •00001 06 + •00000 01 + 'OOOOO 01 Sum + •25859 84 + •11423 73 + ■18767 48 + •01515 22 *'. 21 + 2g. 2i — 2g. 4 — ■OOOO I 75 3 — "OOOOO 51 — '00I08 34 2 — "00024 95 — •O4714 61 + •00000 19 1 — '00940 95 + 'ooooo 12 + •29280 13 + '00080 10 -'03505 90 — •00082 36 -•O0795 42 — '02647 °3 • — 1 + •32065 65 + •03486 41 — •O5402 7i — 'oooio 85 — 2 — "02496 30 + •00468 96 — "00194 98 + 'ooooo 02 —3 — '00056 64 + •00002 20 — '00004 05 —4 — -ooooo 9i + 'ooooo 02 — "OOOOO 07 Sum i + •25039 49 + •03875 35 + •18058 20 -■02577 57 Motion of the Moon 29 Characteristic e /2 k : '■Ar* Values of £ + &. 4 3 2 1 o — 1 — 2 -3 -4 Sum + -ooooo 33 + •00013 97 + -00404 2 1 — •22402 63 + •33808 64 -•32859 84 — •01208 10 — '00024 56 — •ooooo 33 — '22268 31 + ■0001 1 94 — •00148 01 — •05109 26 —•00033 08 — 'ooooo 27 -•05278 68 Values of i. (i 2 ^)i. (V 2 /fc' 2 )s- (V*' 4 )i- (v'W\- 3 2 1 — 1 — 2 -3 + -ooooo 7 + •00167 8 -•38676 9 + •01312 8 + -00008 1 + -00009 7 + •00617 8 + •17825 2 + •02792 3 — •00001 8 + •00000 1 + -ooooo I — 'OOOI2 8 —■07136 -•08335 9 + -02048 7 + '00026 7 + -ooooo 3 + -ooooo 6 + ■00033 3 + •00475 7 + •13082 •ooooo + 'OOOOO 2 Sum -•37187 5 + •21243 3 — ■13409 + •13592 SO Mr. Eenest W. Beown, Theory of the Characteristic e /2 k 2 Values of i. \vVck% W'kk'y,. {rm'k 2 ),,. {■Wl' k'-)i. (WM'h 3 — '00002 — -00004 6 + .00000 68 2 — 'OOOO I 9 — 'OO 1 42 2 — 'OOOOO 1 — '00348 4 + '00048 87 I — - OOOOI I -' 72S3 2 + 'OOOO 1 5 -•19279 + •02689 8 O + •27381 —4IS68 + 'OOIOI — - oi9i9 2 + •08383 2 — I + •59306 6 -•O5074 8 + •30225 — '00469 2 — •26628 1 — 2 + '00284 8 — •OOOO I 6 - '00945 8 — "00006 7 — "OOI2I 51 -3 + 'OOOO 2 4 + 'OOOOO 1 — '00004 3 — 'OOOOO 1 — 'OOOOO 96 Sum + ■86972 - '54042 + '29377 — •22027 — •15628 107. Characteristic k. Values of $. For arguments 2i±4g, 21, 2=o. i. 2i + 4g- 24 -4g. 24. 4 + 'ooooo 44 3 + ■00026 86 + '00002 20 2 + 'ooooo 07 + ■01037 01 + '00183 54 1 + •00005 65 — '12623 98 + '12460 70 + •00288 65 + •02178 99 -3-96849 90 — 1 — ■09817 71 + '00042 43 + '09150 01 — 2 + •01326 43 + •00000 58 + '00077 12 -3 + - oooo8 29 + 'OOOOO 57 -4 Sum + 'ooooo 24 -■08188 38 -•09337 67 -374975 76 Motion of the Moon. 3i Characteristic k*. Values of St + &• i. 21 + 2g. 21 — 2g. 4 — -ooooo 03 ! 3 + -00000 03 - -00003 75 1 2 + -00008 09 + -00059 77 — -ooooo 33 1 \ I + -OI375 2 5 + -ooooo 02 — -01261 91 -•00193 31 ! + 1-96250 00 + •00025 03 + 1-95644 07 + -Q39S 1 5 2 1 1 —1 + -01515 37 —■00427 50 - -01125 33 + -00008 92 1 —2 + -002II 13 — •0000 1 36 — -00040 44 + -ooooo 04 ! -3 + -00007 78 — -ooooo 01 — -ooooo 64 -4 1 + "00000 15 i Sum + 1-99367 80 — ■00403 82 + 1-93271 74 + •03766 84 Values of i. (*% + -oooo 1 3 J (***')(. (**"),. C^/t"),-. 3 — -ooooo 6 + •00000 05 2 + •00072 4 — •00053 8 + -00002 68 1 — -05111 5 — •00003 6 — •03284 — -OOOI2 O + •00001 1 + •00171 4 — •00096 1 + •98818 1 — •99206 I — 1 + -00029 6 + •00001 5 + ■02794 — •001 2 1 8 + •06300 2 — 2 + •00412 + •00260 9 — •0000 1 5 + -00004 69 -3 + -00004 2 + -00446 9 + •00000 9 + -ooooo 02 Sum — -04864 9 + •02956 + '9535 6 -•92910 5 32 Mr. Ernest W. Brown, Theory of the 108. Characteristic e 8 ct. Values of &+&. 2.1. 21 + 3?- 21- 3c- 9 — -00018 71 7 + "OOOOO 19 — '01 000 07 — "OOOOO 34 5 — '0O0O2 09 -•09655 71 + 'OOOOO 30 3 — -00939 13 — 'OOOOO 27 + -04866 . 76 + '00008 93 i — -63721 77 — -00036 5 2 + •18169 35 — '00037 34 — i + 1-05819 96 + -00042 40 -•05478 44 + •00181 04 -3 + -05883 14 + -00032 86 — '00065 86 — '00181 00 -5 — -02176 80 + -00035 12 + 'OOOOO 19 — '00003 19 -7 — -00114 07 + -oooo I 62 — 'OOOOO 04 -9 Sum — "00001 96 — - 0000 1 67 + '44747 47 • + ■00073 54 + •06817 5i — '00031 64 2(. 2.i-i - c. 2J- -c. 9 + 'OOOOO 04 — '00002 °5 7 — '00005 87 — '00220 26 — 'OOOOO 03 5 — 00826 16 — -ooooo 17 — 'I2I22 64 — '00004 70 3 - -48587 63 — -00023 37 ' — '64460 95 — 'OOOO I 98 1 - 71097 95 + ■00184 02 + ■18169 80 — •00102 75 — 1 + '19813 19 — '00148 48 —•15240 58 — ■00236 47 -3 - -13854 40 — '00165 65 — '04646 56 + '00442 62 -5 — '01271 97 + •00110 69 — '00069 53 — '00109 85 -7 — '00020 42 — 'O0O2 2 96 — 'OOOOO 45 — 'OOOOI 93 -9 — 'OOOOO 24 — 'OOOOO — '00066 40 32 — 'OOOOO -•78593 °3 — 'OOOOO 03 Sum —1-15851 41 2 S — '00015 12 Motion of the Moon. 33 Characteristic e 3 a. Values of 9 7 5 3 i — i -3 -s -7 -9 Sum («•«)«■ — - ooooo 3 — -00027 4 — -01911 6 + •07964 1 + -03460 —•00230 3 — -00002 7 — 'OOOOO I + •09252 («"a) f — 'ooooo s — "00039 8 —•00925 5 — -01426 + •01030 2 — '00130 2 — -00005 9 — -01498 («*«'«),. — 'OOOOO 2 — '00023 9 — -01591 3 —•05442 6 + •18382 — -01911 2 — -00023 8 — -ooooi o + •09388 — "ooooo — '00006 —•00436 -•05569 — -10225 — -02624 — -00089 — -00004 -•18955 109. Characteristic eVa. Values of .£ + &. 2i, 21 + 2C + m. 2i — 2C- -m. 2i' + 2C- -m. 9 — "00009 7 + -00003 — •00668 + 'O0004 5 + -00262 — •29203 — -00027 + 'OOO36 3 + -16029 -•2759O + •00137 - - °3993 — •00004 1 + 6-03740 — -ooooi -•"737 — "00649 - 77599 — "00262 — 1 — +2403 + •00261 + •74524 + "00422 -1-83937 + •00191 3 — -12488 —•04085 + •01684 + •00599 + -15188 + •00185 -5 - -045 r 5 + •00835 + •00025 + '00009 + -03582 — -00200 ~7 — -00078 —•00159 + '00071 + '00028 -9 Sum + -ooooi — '00003 + '0O002 + 5' 6 °55i —•03152 + •07026 + •00491 — 2+6646 — '00062 Royal Astkon. Soc, Vol. LIV. 34 Mr. Erxest- W. Brown, Theory of the Characteristic e 2 e f a. Values of «+ 8. 2i. 2J - 2C + m. : 2t + m. 2J- -m. 9 + - OOOI2 + '00002 + *0OOO2 7 + -00688 + '00115 — 'OOO23 5 + •26155 + 'OOOOI + '06615 -•O3842 — '00002 3 + •39707 + '00047 + 2-47134 + •00001 — 1-40678 — '00179 i — '62492 -■00995 + ■89452 + "00425 — ■69566 + •00653 — i — •IOO50 + '01256 — •27015 -'°3575 + '3°3° 8 — •01 102 -3 — '00121 -•01973 — "20414 + -04986 + •32001 —•00456 -5 + 'OOOI2 — "00042 — •O0349 — "01142 + •00706 + •00259 -7 -9 + 'OOOOI — 'OOOOI — '00002 — "00021 + •00010 + '00004 Sum -•06088 -•01707 + 2-95538 + •00674 — 1 - 5 1082 — '00823 Values of 2/, (e^aV;. («'y«)<. ( E y°)<. (yv);. («'1«)i. («Y°)f. 1 7 — '0002 + '0003 5 -•0177 +•0129 + '0023 — '0013 1 3 + '005 I -•2686 — 'OOII —■0879 + '1281 — 'P7 46 1 + •2742 -•1039 -•0435 +2T242 + 1 '446 1 — •0768 .; — 1 — '0294 +•0341 — 1-0316 — 'OOIO -' I 597 -3-8796 ] -3 + '4801 + '0005 + •4427 — '0009 — '0009 + " OI 35 | -5 — '0003 +•0013 — '0007 + '0002 1 -7 — '0002 — "OOOI ' 1 1 Sum + 729S -•3558 —•6323 + 2-0476 + 1+152 — 4-0186 . 1 Motion of the Moon. 35 no. Characteristic ee' 2 ct. Values of j? + i\ 2i. 1 9 2 i + c + 2m. 7 — -00004 4 5 — •00287 6 3 -•13497 9 4--ooooo 7 i — 2-50990 8 — "00292 — i — •80061 9 4- -01047 1 -3 -•15688 6 + '34620 1 -s — •00452 7 — •05404 8 -7 4- - ooo 1 1 7 — •00079 1 -9 — •0000 1 Sum -3-60972 2 4-'2989i 21 — e — 2m. 4- 'oooo 1 1 4- '00026 6 — ■04866 o —'00005 2 — •76647 7 — "00666 6 — •62264 6 4- '03178 7 — •45296 4 4- '01308 5 — -01619 4 —'00036 4 — •00031 5 —'00003 7 — 'ooooo 2 — 'OOOOO 1 ■ 1-90698 1 4- '03775 2 24 + C -2m. 4- -oooi 7 4- -00424 I — -ooooo 2 — -0220I 2 —"•00024 5 4- '26600 8 — -01207 2 — -47218 9 + '01051 8 -•06595 8 4- -00408 9 -•00350 9 + •00233 8 ■•00007 7 + -ooooo 5 -•29332 6 +'00463 1 2(. 9 2i — C+ 2m. 7 — 'OOo6l O 5 — •O2829 2 — 'OOOOO I 3 -'35775 3 — •00013 I ! J — -52080 9 + •00478 9 i - 1 4- '06614 8 4- -09080 3 ! -3 4- 'O0826 5 — •13086 1 1 -5 4-'ooo92 1 — •00267 4 ! —7 4- '00003 6 — -00003 7 ! -9 — -ooooo I 4- -00023 9 4- -01488 5 + •64255 4 4-2-14351 8 + 73440 6 + •61312 7 4- -02246 9 4- '0004 1 5 4- 'ooooo 3 4- 'OOOOO I 4-'oooio 7 — '01498 3 + '03484 2 — -11872 9 4--oi657 8 4- -00023 1 4- -ooooo 3 4- -00004 6 4- -00360 1 4- -17806 3 + 2'9i3+5 5 + •68113 3 4- "40067 2 4- -06910 6 + •00143 3 4- -00003 4 — 'ooooo 3 4- '00266 3 + "01973 I 4- '02618 8 4- -04654 8 4- '00087 ° 4- '0000 1 2 Sum -•83209 4 --03811 3 4-4-17161 6 —-08195 o -4-24754 3 +-09600 9 2,6 Mr. Ernest W. Brown, Theory of the Characteristic ee^a. Values of I 21. (*V 2 a)i. («YM«. («? ,! <0«. («V<0<- («Il'a)j. J («W«) 4 . 5 — •0023 — - OOII + '0004 + •0071 3 — •0050 — •1649 + 'OOOI — •0297 + •0211 + •2706 i -•l86 S — '1208 + -005 1 — "1912 + •1708 — •0926 — i — "2071 — •0297 — •0445 + •1006 + "4363 + "0461 -3 + •4312 — '0004 — '0004 — -0126 — •0316 + •0067 -5 — - oo6i + •0001 — •0001 + "0023 + •0001 Sum + '0265 -•3181 — •0396 — '1341 + '5993 + •2380 in. Characteristic e' l a. Values of £ + £. 2i. 2i + 3m. 25- 3m. 9 + •00001 4 7 + '00000 9 + •00037 I 5 + -00043 4 + •00820 4 — 'OOOOO 4 3 + •00474 3 + '00006 4 -■19909 9 — "00062 1 —•29874 9 — •00700 1 — •04918 3 -•03895 6 — 1 — '00992 9 + "°5°54 9 — •09107 7 + '00285 -3 -■03587 3 — •60765 5 + •00026 9 — "00063 4 -s + "00204 6 — •00855 2 + '00004 9 — "OOOOO 8 -7 + '00006 6 — 'OOOIO + 'OOOOO 1 -9 + "OOOOO 2 — -ooooo I Sum -•33725 1 -•57269 6 -'3304s 1 ~'°3737 2 Motion of the Moon. 37 Characteristic e' 3 a. Values of Jt+g. 2%. 2i + m. 2i- m. 9 + - 00000 9 7 — '00015 9 + '00040 8 S —•00858 7 + '02460 2 + 'OOOOO I 3 — •26123 8 — '00002 3 + -83332 + '00004 5 i -•01739 6 — '00566 6 + -25130 2 — •00662 8 — i + •08868 9 -•03773 4 — '00639 3 + •01028 3 -3 + - i43°S 5 + •31427 9 — '04100 1 —•00738 4 -5 + •00288 4 + '00347 8 — '00104 5 — -00021 -7 + '00004 4 + '00004 — 'OOOOI 8 — 'OOOOO 4 -9 + '00000 1 + '00000 1 Sum — •05270 7 +•27437 5 + i'o6n8 4 — •00389 7 Values of 21. (^)i. 0?"«)i. 0?Y«)<. (rm'*«U- 5 + •00001 + -00036 — '00028 + •00072 3 — •00012 + •00972 — -01429 + •03354 1 — •0723 + •3042 -•3287 -•2531 — 1 -•7294 + •0898 + •7869 + •9756 -3 -•1709 — '00002 + •08589 —•00436 -s — -00046 + •00028 — -00004 Sum -•9732 + +04I + •5298 + •7524 38 Mr. Ernest W. Brown, Theory of the 112. Characteristic elcos. Values of jt + g. 21. 2i+C + 2g. 2! -e-2g. 9 — 'OO002 63 7 — '00000 06 — -00232 93 5 + 'OOOOO J 3 — •12694 76 — -00003 62 3 — '00042 98 + -ooooo 01 + •18390 29 — •00016 14 i - -05518 61 + •00002 60 + •23086 82 + •00590 31 — i + 1-98177 62 — -00800 76 —•54487 76 —•00573 30 -3 + '21544 85 + •01729 16 —•00453 24 — '00009 3 1 -s — -03239 84 — •00025 38 + '00001 95 — "OOOOO 10 -7 — -00064 34 — -00017 96 + 'OOOOO 06 -9 — -ooooo 68 — -ooooo 19 Sum + 2-10856 09 + •00887 48 — •26392 20 — 'OOOI2 l6 zi. 2i + e- -2g. 21- -C+2g. 9 — 'OOOOO 09 7 — '000 1 3 36 — 'OOOOO 25 5 — '01084 86 — 'OOOOO *5 — '00030 68 3 -•51891 28 + 'OOOI4 81 — '01792 58 — 'OOOOO 56 I -•14053 5i — •OO594 66 + 170964 27 + '00054 14 — I + '54234 01 + -OI3IO 49 - -18777 01 + •02116 52 -3 -•14735 54 + - OII24 11 — '11929 11 —■02572 49 -5 — '00168 14 + '00004 72 — '00276 70 + •06047 81 -7 — 'ooooo 68 + 'OOOOO 02 — '00003 48 — 'OOOOO 26 —9 — 'OOOOO 01 — 'OOOOO 01 Sum —•27713 45 + •01859 34 + 1-38154 45 —•00354 85 .Motion of the Moon, 39 Characteristic ek 2 a. Values of jt + g. 21. 2i + C. 21 - C. 9 + -ooooo 34 7 + '00001 07 + -00023 93 + -ooooo 03 5 + - OOI26 86 — -ooooo 10 + -02399 07 + -00002 16 3 + - U95 8 89 — -00009 19 + 1-50798 22 — -00409 44 i + 675069 59 + •01948 61 -178455 48 — -02I57 33 — i -1-93792 53 — -04226 85 + 1-07322 87 — •O2465 41 —3 + -19026 33 — ■02071 99 — -02962 9i + •00703 99 -5 — -00889 43 + •00496 60 — -00036 56 + -OOOI3 23 -7 — -OOOI2 71 + -00004 56 + -ooooo 03 + -ooooo 15 -9 — -ooooo 10 + -ooooo 04 Sum + 5-11487 97 -•03858 32 + -79089 5 1 —•04312 62 Values of 2/. («**■).-. («'A' 2 o)i. (e£'-a);. (e'Fa\-. 7 — -OOOI2 7 — -ooooo 5 5 — -01181 1 — -00052 + -ooooo 4 3 + •0000 2 3 — •08718 3 — -04041 2 + •00064 2 1 + •00187 9 — •O3664 1 — '2022O + •11156 3 — 1 + •15508 4 — -04612 8 + •17870 T +•10583 -3 + •15427 2 — •OOOI7 7 — •OI258 5 —•04372 5 -5 — ■00719 — '00006 3 — -00007 6 -7 — -00005 1 — -ooooo 1 — -ooooo 1 Sum + •30401 7 — •18206 7 — -07709 + ■17424 40 Mr. Ernest W. Brown, Theory of the Characteristic ek 2 a. Values of 2J. («/M'a)i. ^kk'a) t . 7 5 3 i — i -3 -5 -7 4- "ooooo i 4- "oooo6 3 4- '00669 3 + -56588 5 — 2-22530 — -10800 4 — -00080 9 — "ooooo 5 4- -ooooi s 4- "00166 1 4- "16219 6 + i"44U3 — -03941 i — -00373 9 — "ooooi 5 Sum — 1-76148 4-1-56184 1 1 3. Characteristic e'k 2 a. Values of & 4- g. 21. 2i + m + 2g. 2( — m - -2g- 9 — -ooooo 71 7 4- "ooooo 18 — -00049 35 5 4- "00017 33 - -03515 46 — -ooooo 07 3 4- -01544 48 4- 'ooooo 02 — -07602 70 — •00097 56 1 4-1-06559 78 4- "00041 62 — '28776 46 -•00955 78 — 1 - -60593 45 -•04399 71 4-3-05205 66 — -01139 56 -3 4- -00478 °7 — "ii86o 46 +' °5553 97 4- -00002 02 -5 - -01154 32 — •00649 08 4- "00070 44 4- -ooooo 06 -7 — -00014 63 — •00006 41 4- "ooooo 78 -9 — -ooooo 21 — •ooooo 05 Sum + -46837 23 -•16874 07 4-2-70886 17 — •02190 89 1 Motion of the Moon. 4i Characteristic e'k^a. Values of M + S. 2i. 2i + m- -2g. 2i — m + 2g. 9 + 'OOOOO 20 7 + '00021 14 + -ooooo 18 S + •01621 97 — "OOOOO 34 + -00007 23 3 + •49466 56 + -00069 13 + -00293 93 + 'OOOOO 05 1 —•97149 67 -•00578 5° + -07585 28 — -00011 26 — 1 — '02020 43 —•04771 S3 — 2-92221 25 -•01749 37 -3 + "00345 46 — •00060 99 + -18175 84 + •02886 88 -5 + •00035 14 — -ooooo 62 + -00704. 5o + •00308 27 -7 + 'OOOCO 9i + -00009 40 + •00001 74 -9 + 'OOOOO 02 + 'OOOOO 06 + -ooooo 01 Sum -•47678 70 -•05342 S5 —2-65444 83 + •01436 3 2 21. 2i + m. 2t — m. 9 — 'OOOOO 04 — -ooooo 10 7 — -ooooi 65 + -00003 17 5 — '00150 99 + -00417 5 2 + -ooooo 02 3 — -09289 49 + •00002 48 + -33260 II + -00008 40 1 -1-26453 15 — •00687 78 + 1-00154 29 + •01019 64 — 1 + -43988 60 + •19119 70 - -44129 44 + ■02820 24 ~3 - -08255 25 + •11614 20 + -19227 95 —•03913 82 -5 — -00097 39 + •00120 59 + -00317 00 — •00025 30 -7 + -ooooo 62 + '00001 11 + -00003 92 — 'OOOOO 19 -9 — 'OOOOO 02 + -ooooo 04 1 Sum — i - oo258 76 + •30170 30 + 1-09254 46 — •00091 01 Royal Astron. Soc, Vol. LIV. q 42 Mr. Ernest W. Brown. Theory of the Characteristic e'k 2 a. Values of 2J. (>)£-<«);. (V'k"a) t . („£'%);. (l'A 2 «)<- 7 5 3 i — i -3 ~5 -7 + -OC004 4 + •00358 8 + •10476 ■+'744i7 — •00328 7 — 'OOOOI I — '00002 9 — •OO335 8 -•34996 -•68238 + '50351 5 + •00284 3 + •00001 9 + •00001 I + •00142 I + •17656 -•02358 -•01345 I 4- -00013 7 + - ooooo 9 + 'OOOOO I + '00008 6 + '00346 5 — "48005 -•20583 + ■00157 1 + - ooooo 5 Sum +,•84926 J --52935 + ■14111 -•68075 2i (#'a) f . (I'M'aV. 7 5 3 1 — 1 ~3 -s -7 — "OOOOO 2 — -00026 7 — '03252 O — 4'3 r 577 — 1-32620 — -00148 2 — -ooooi 7 + -OOOOO 2 + -00022 O + "02120 2 + 7I84I + I2"282l6 + -03487 2 + -OOOI6 4 + "OOOOO I Sum — 5-67626 + i3'°57°3 Motion of the Moon. 43 1 1 4. Characteristic e 2 a fin* Values of jt + g. i. 2t + 2C. 2i- -2C. 22. 4 + •00028 + '00007 3 + - 000l8 + ■01627 — - O0OO4 + '00281 2 + •00754 + •29412 -•OOO33 + -12784 — •OOO35 1 1 + •26891 — •00061 — •12528 -•OO745 + ro2879 + •00473 -•37623 + •03189 — •06085 + - ooo66 - - 3°835 - -00958 — 1 — T4252 — •01204 + •02427 + '00084 + ' 2I 573 — •OI47O — 2 + •06237 -•03041 + ■00086 — '00632 + '012 20 + •02769 -3 + •00174 + ■00495 + '00003 — '00008 + '00031 -•OO245 -4 + -00004 — •00024 + 'OOOO I — -00003 Sum -•17797 — •00646 + •14970 —•01272 + I-0794I + •00531 Values of i. («•<•'),. (/-a-),. («'« 5 )f. 3 + •00069 + -OOOOS 2 + •00018 + "033° + '00442 1 + •00829 + •035 + '08709 — •02731 — •0025 -•079S — 1 — "°95 + •00077 + •0172 — 2 — •0249 — -00020 + •00291 -3 + •00054 — '00008 Sum -•138 + •067 + •0321 44 Mr. Ernest W. Brown, Theory of the 115. Characteristic eeV. Values of ,)? + £. i. 2J + C + D1. 2i— 2 — m. 4 + -0002 I 3 — '00009 + ■00753 — •0000 1 2 . — '02560 + •26215 — •00263 1 -2-75577 — -00278 + •20795 + -00899 + 1-34099 — •00605 + •41844 + •00925 — 1 + -08386 + •01829 -•34740 + •01662 — 2 + "045 ro + •24946 — "00299 + '00849 -3 + -00099 — -02085 — "00002 + '00007 -4 + -00003 — ■00024 Sum -1-31049 + •23783 + •54587 + •04078 i. 2» + — m. 2i — c + m. 4 + -00003 — -O0O06 3 + •00125 - '°0753 2 + '03440 — -00003 — '72035 — •OOO53 1 + •23283 — -00644 — I-I360O + - 0053I + •61656 + •02227 + -22661 + '02298 — i -•40173 + •00163 + '03323 + •05310 — 2 — -10185 —•05084 + -00479 -•07805 -3 — -ooioo + •00181 + -0002I — -OOI02 -4 — -ooooi + 'OO002 + -ooooi — 'OOOOI Sum + -38048 -•03158 -1-59909 + -OOI78 Motion of the Moon. 45 Characteristic ee'a 2 . Values of i. (*ya'% (Wh («)'<«% ( e 'iOi- 3 2 I O — I — 2 -3 — '00001 — '00131 —•12949 + ■1225 —•1899 + •0662 — "00I2I + •00026 +•00751 + •1213 + -275 1 —•01338 + •00047 + •00003 + •00106 + •01164 + •2975 — •2103 -•01635 + •00010 —•00037 -•03513 — '1270 -•5059 + •01300 — '00417 — '00002 Sum -•1332 + '39!3 +'0837 -•6596 116. Characteristic k s a 2 . Values of J? + S. i. 2! + 2g. 2X- -2g. 2J. 3 + •00001 + •00037 + "00004 2 + •00047 + -i53 l6 + "00009 - 'O0055 + -00002 1 + •01697 + •00027 — -22860 + •00117 — 1-50841 — -00l80 -•46493 -•03704 —•07083 — '10214 + 74163 + •10091 — 1 -•23351 — -03121 + •08806 — -02647 — "10082 + T3475 — 2 + •04289 -•03335 + •00279 — - ooo 1 8 + "00400 + - 02l64 -3 + ■00017 — •00009 + •00005 + '00012 + •00014 Sum -•63793 — T0142 — •05500 —•12753 - "86399 + - 25566 Values of .-. (W) 4 . (*«««),. (K-V);. 3 + •00015 — 'OOOOI 2 + •02250 — '00110 1 + •00008 + •632 — -17168 -■03354 — •0508 + •2109 — 1 — •623 + •00535 +•2299 ! — 2 — -0300 + -00009 + •00274 ! ~ 3 — '00002 + •00001 j Sum -•686 + •609 + •2708 46 Mr. Ernest W. Brown, Theory of the 117. Characteristic ke 3 . Values of K+L. For arguments 2t'+g±2c, L=o. i. 2J + g+3C. 2i + g-3C. 5 + •00000 56 4 + •00000 03 + •00022 74 3 + "ooooi 44 + 'o°S33 39 2 + •00074 35 + •00402 52 1 + •03004 59 — "01541 20 + •71696 43 — - o75°3 40 — 1 — - ioo8i 77 —•02229 53 — 2 + •00022 10 — '00089 °° -3 — '00021 24 — '00002 15 —4 — -00015 2 8 — -ooooo 03 -5 — '00000 65 Sum + '64680 00 — '10406 10 i. 2i+g + C. 2i + g-e. s + 'OOOOO OI + 'OOOOO 16 4 + •00000 83 + '00008 52 3 + -00043 36 + •00340 21 2 + •01736 So — -OOOOO 07 + •07930 55 — "OOOOO 12 1 + •40822 H — -00007 16 + •00652 77 — - OOOII 17 -•01854 4i —•00635 S3 + •06525 69 + •00001 00 — 1 + •00110 98 + •00022 09 — '02001 07 — 'OOOOI 61 — 2 — '00204 37 — '00003 89 — 'OI2I I 33 — 'OOOOO 02 -3 — '00229 06 — 'OOOOO 04 — '00050 44 — 4 — '00009 ■ 7 '6 — 'OOOOI 26 — 5 — 'OOOOO 25 — 'OOOOO 02 Sum +•40415 97 — '00624 60 + -I2I93 78 — •0001 1 92 Motion of the Moon. 47 Characteristic ke 3 . Values of i. <*•'),- - — (lce' 3 Y. (*«'«% (W) r 4 + 'OOOOO 70 + •00000 01 + -ooooo 14 3 + "00000 01 + •00038 82 + 'OOOOO 68 + '00009 92 2 + •00001 12 + •00180 89 + '00048 '5 + •00531 28 I + ■00079 37 + '01349 47 + •02534 49 + •00201 07 O + ■04139 88 — '02230 57 — '00708 02 -•°SS35 33 — I —■02439 3i — '00142 20 — '00106 86 — •00749 18 — 2 + ■00007 65 — "00002 5o — '00094 57 — -00083 02 -3 — 'OOOI2 54 — 'OOOOO °3 — '00016 94 — •00001 49 —4 — 'OOOOI 23 — 'OOOOO 3° — 'OOOOO 02 Sum + •01774 95 — -00805 42 + •01656 64 — •05626 63 118. Characteristic ke 2 e'. Values of K+L. i. 5 2t+g + 2C + m. 2i + g — 2C — m. + 'OOOOO 66 4 — 'OOOOO 04 + '00030 69 3 — '00002 30 + •01032 92 + 'OOOOO °5 2 — '00107 37 + 'OOOOO 01 + •16457 95 + '00004 19 1 -'03569 68 + 00001 10 — •01782 22 + •00016 40 -•5456o 36 + •00091 97 + •00562 27 — "00079 83 — 1 — •05701 99 —•00043 15 + '04641 16 — '00003 02 — 2 + •00702 83 — 'OOOOI 58 + •00202 05 — "OOOOO 03 3 — '00663 19 — 'OOOOO 13 + '00005 09 t — 4 — '00036 53 + 'OOOOO 09 -5 — 'OOOOI 01 Sum -•63939 64 +•00048 22 + •21150 66 — -00062 24 48 Mr. Ernest W. Brown, Theory of the Characteristic keV. Values of K+L. i. 2t + g+2C- -m. 2t + g- -2c + m. 5 —•00000 08 4 + •00000 26 — -00004 42 3 + 'OOOI2 68 — "00149 82 + "OOOOO 05 2 + •00539 69 + "OOOOO 01 — •02391 83 + -00004 19 I +■14580 39 + •00001 10 — '00160 56 + -00016 40 O +•65773 97 + -0009 1 97 + •00971 18 — -00079 83 — I —•00337 70 —•00043 15 -•08389 85 — '00003 02 — 2 + "00043 36 — -ooooi 58 — '00626 45 — -ooooo 03 —3 + •00177 32 — 'OOOOO 13 — - oooi9 66 —4 + -00007 97 — "00000 40 -s + - ooooo 20 Sum + •80798 14 + '00048 22 — '10771 89 — "00062 24 i. 2i + g + m. 2i + g— m. 5 — -ooooo 02 + ■ooooo 12 4 — -ooooo 91 + •00005 78 3 —•00043 65 + ■00260 OI 2 — -01468 °9 + "OOOOO 47 + •08054 40 + 'OOOOO 29 1 -•23463 84 + "00040 37 + •96859 73 + •00025 48 — -02250 64 + ■00011 90 — •01639 85 — 'OOOI2 32 — 1 + •01512 91 — "00065 5o — •00262 58 + •00018 93 — 2 —■04967 40 — "OOOOI 82 + •01797 53 — "OOOOI 06 -3 —•00305 52 — "OOOOO 02 + •00079 78 — "OOOOO 01 —4 — '00009 00 + ■00002 02 -5 — "OOOOO 17 + •ooooo 01 Sum -•30996 33 — "00014 60 + 1 •05156 95 + ■00031 31 Motion of the Moon. 49 Characteristic ke'Y. Values of i. (£e 2 7))i- (^V)i (A«Y)i (*<='%,)« 4 + 'OOOOO 69 — 'OOOOO 09 3 — 'OOOOO 03 + '00047 35 + '00000 16 — •00006 38 2 — '00002 10 + ■02356 09 + 'OO01O 98 — '00297 65 I —•00134 79 + -03846 7 + •00579 68 + •01507 7 O -•05587 76 + •04524 8 +•07543 81 — •04015 9 — I — •10870 Si + •00541 35 — - 02o6l 5 — '01098 67 — 2 — '00801 92 + '00008 3° — '00037 8 — '00027 16 — 3 — •001 1 I 3 2 + 'OOOOO 1 1 + '00025 36 — 'OOOOO 42 — 4 — "0000 1 33 + 'OOOOO 37 — 'OOOOO 01 Sum -•17510 26 + '"3 2 5 4 + '06061 1 -•03938 6 i. (fe'7i)i (X-eeV) 4 — 'OOOOO 01 + 'OOOOO 07 3 — 'OOOOO 89 + '00005 55 2 — '00058 9 1 + '00344 09 1 — •02666 26 + '12360 04 — '12700 5 + '08781 9 — 1 — '02647 80 + •00687 6 — 2 —■00730 21 + •00231 70 ~3 — •00014 20 + '00003 48 —4 J Sum — 'OOOOO 20 + 'OOOOO O^ — -18819 + '22414 5 liOYAL ASTROX SOC, VOL. LIV If 5° Mr. Eenest W. Brown, Theory of the 119. Characteristic. kee' z . Values of K+L. 1. 2! + g+C- -2m. 2i + g— c — 2m, 5 + 'OOOOO 21 4 + 'OOOOO 02 + •00012 17 3 + 'OOOOO 72 + ■00499 31 + 'OOOOO 02 2 + '00020 64 — 'OOOOO OI + •12823 94 + '00002 61 1 + •00011 43 + 'OOOOO 86 + '78746 00 + ■00234 71 — •17808 97 + •00295 39 -■17852 33 — •O0867 68 — 1 + 71766 93 — -00484 °3 + '01447 01 — 'OOOOO 22 — 2 — '05048 72 -. — '00012 45 — '00026 52 + 'OOOOO 02 -3 -•00595 Si — 'OOOOO 10 — ' 0000 1 76 —4 — '00020 16 — 'OOOOO 04 -5 — 'OOOOO 48 Sum + •48325 90 — '00200 34 + 7S 6 47 99 — -00630 54 i. 28 + g + C - -2m. 28 + g- -C + 2H1. 5 + 'OOOOO 02 4 + •00000 94 + 'OOOOO 01 3 + '00046 12 — '00006 97 2 + •01699 66 + 'OOOOO 10 — '00721 63 + 'OOOOO 16 i + •33938 80 + •00009 10 -•35 J 59 54 + '0006l 65 + '3°9 I 3 00 + •00473 20 + '31108 88 — •01333 33 — 1 — •33037 02 — '00144 85 -•12455 46 — '00037 07 — 2 + •00733 70 + 'OOOOO 07 — •01967 10 — "OOOOO 40 —3 + '00010 3i + 'OOOOO 01 — -00075 20 —4 — 'OOOOO °5 — '0000 1 81 -5 — 'OOOOO 01 — 'OOOOO 01 Sum + •34305 47 + ■00337 63 — '19278 83 — "01308 99 Motion of the Moon. Kl Characteristic \ee'' 2 . Values of K+L. '<■ i 5 2i + g + e. — 'OOOOO 21 + g- -c. - - ! 05 4 — 'OOOOO 29 — '00003 45 — ■00014 54 — '00140 70 — 'OOOOO 08 i 2 — •00554 54 + •00000 44 —■03566 80 — '00007 79 | I — -11838 75 + •00042 44 — -20612 34 — •00712 07 O — •05326 65 + •03416 3 2 -■03830 00 — •00675 3 2 — I — •13926 03 — '00502 70 + •10602 30 —•00151 32 ; — 2 + •03413 5 2 + •00010 36 + •01041 88 — 'OOOOl 72 1 1 -3 + •00264 53 + 'OOOOO 13 + -00033 81 — 'OOOOO 01 -4 + '00007 99 + 'OOOOO 72 -s Sum + 'OOOOO 16 -•27974 60 + •02966 99 —•16474 63 —•01548 J' Values of 4 3 2 1 o — 1 — 2 ■■> j — 4 Sum (*«.*);. + •00000 01 + •00000 52 — -00002 93 •04371 -,6 — -66115 99 -•03574 62 — •00052 45 — 'OOOOO 69 — -74117 71 (*«V)i. + 'OOOOO 2 I + •00015 6l + •00971 71 + '35 1 73 9i + •15803 06 + •00480 33 — '0000 1 15 — 'OOOOO o^ + •52443 (/.■e„' 2 ) ; . + ■00000 OI + ■00000 75 + •00049 78 + •02306 46 + ■11337 20 + •28041 72 + •00383 22 + •00001 26 + •42120 40 (/«V;j — 'OOOOO 1 24 — •00052 32 J — '09968 00 -■25777 09 -•0595 8 97 — •00152 II : — '00002 38 s — "OOOOO 03 -•41911 14 52 Mr. Ernest W. Brown, Theory of the Characteristic kee' 2 . Values of i. (hy<i')t. (fe'vv'h. — 'OOOOO 06 4 3 — 'ooooo 23 — '00004 14 2 — •00015 34 — •00245 63 I — "00732 98 -•07397 30 — •00507 19 + •03863 62 — I + •12496 93 +•03793 55 — 2 + •01613 76 + •00073 98 3 + '00021 08 + '00001 02 1 —4 + 'OOOOO 26 + '00000 01 1 Sum + •12876 29 + •00085 05 1 20. Characteristic ke' 3 . Values of K+L. i. 2i + g + 3m. 2i' + g- -3m. 5 + 'OOOOO °3 4 + 'OOOOI Si 3 — ooooo 03 + '00070 49 2 + ooooo 03 + '02390 12 + 'OOOOO 36 1 — 00016 08 + '0000 1 88 + ■39028 72 + '00034 06 1 — 01185 45 + '02137 21 — "02187 35 + '02201 69 \ — 1 + 39014 38 — "00489 89 — '00064 00 + '00003 96 1 i 2 — 02391 °s — '00002 03 — '00019 99 + 'OOOOO 02 ! _ 3 — 00120 16 — 'OOOOO 01 — 'OOOOO 23 j —4 — 00003 10 ! S — ooooo 05 I Sum + 35298 49 + ■01647 16 + •39219 30 + '02240 09 Motion of the Moon. i>S Characteristic ke' 3 Values of K+L. i. . 2! + g + m. 21 + g-m. . 5 4 + "OOOOO °5 — -ooooo 64 j + '00002 55 — '00029 19 2 + •00062 54 + -ooooo OS — "00972 22 — "OOOOO 04 I + •00128 4t + '00008 89 —•15175 42 — -ooooo 49 O -•00357 95 + •01132 25 + •00011 85 + •01084 69 — I -•14427 67 — ■00318 17 + -00047 58 + -00029 32 — 2 + •01680 23 — "0O002 14 — •00094 04 + -ooooo 45 -3 + '0007 T 34 — 'OOOOO 01 — '00006 78 + -ooooo 01 -4 + •00001 71 — -ooooo 20 -5 + -ooooo °3 Sum -•12838 76 + ■00820 87 — -16219 06 + •01113 94 Values of i. '. Oj :, ),> (/b)' 3 ),:. (AW) ■ (Aw' 2 );- 4 + 'OOOOO 02 — -OOOOO 01 3 + •00001 59 + -ooooo 05 — 'OOOOO 63 2 + •00110 53 + '00002 s° — "00041 80 I — '0000 1 49 + •05659 12 + •00025 06 — "01944 00 + •00482 66 — •00172 9 + •06378 I —■06330 2 — I —•53057 50 — '00381 2 + •30385 2 3 + '00018 — 2 -•00513 58 — '00002 82 + '00300 92 — 'OOOI2 08 -3 — '00006 46 — 'OOOOO 01 + '00003 56 — "OOOOO 29 —4 — 'OOOOO oS + 'OOOOO 04 Sum -•53096 45 + •05214 3 +'37095 5 — "08311 54 Mr. Ernest W. Brown, Theory of the 121. Characteristic k 3 e. Values of K+L. For arguments 2i + ^g±c, L—o. i. 21 + 3g + c. 2( + 3g-e. 5 4 I 3 + 'ooooo 04 — 'ooooo 43 2 + '00001 74 — '00041 18 I + •00087 34 — "03237 9° O + •02556 39 — 1 '65684 02 — I — •27846 00 — '11611 13 — 2 — •00185 73 + '00073 33 3 — '00022 02 — - oooor 73 —4 — "ooooi 68 — 'OOOOO 10 -5 — - ooooo 06 Sum — •25409 98 — 1-80503 16 | i. 5 _ . 4 — 'OOOOO 02 3 — '0000 2 16 2 — '00 1 00 69 1 — '02309 88 +•86505 °s - 1 -•09633 83 — 2 —•00055 86 3 — '00016 69 -4 — 'OOOOO 5 2 — 5 + "OOOOO or 26 + g + C. 2J + g-C. — 'OOOOO 65 — 'OOOOO 01 — '00036 08 + 'OOOOO 16 — 'OOOOO 57 — '01471 73 + •00015 30 —•00055 81 - -24570 64 + '01382 37 -'°4955 90 + 2-50685 95 — '00124 00 + •00172 26 + -10206 4i + •00199 63 — '00030 34 + -00127 02 + •00002 21 — 'OOOOO 33 + 'OOOOO 58 + -ooooo 02 + '743 s 5 4i — -04870 + 2-34940 86 + •01475 6 9 Motion of the Moon. 55 (k 3 e);. Characteristic k 3 e. Values of (*•=*'«),. (A-V/e'},- 4 — 'OOOOO 01 3 — 'OOOOO 01 — -ooooo 03 — -OOOOI II 2 + •00000 °3 — -00001 35 — -00002 72 — '00106 94 I + •00002 27 — •00233 85 — •00128 45 — -08994 42 O + •00140 98 — ■46095 5 2 + ■23505 59 — 2-12659 97 — I — •06570 41 + •10654 96 + ■07945 50 + -02789 93 — 2 + •00250 93 + •00095 26 + •00007 65 + -00009 36 3 — - OOOI2 15 + •00000 06 — 'OOOOI 17 + -ooooo 02 — 4 — 'OOOOO — ■06188 15 5o 45 — 'OOOOO +•31326 02 35 Sum -•35580 — 2-18963 J4 122. Characteristic kY Values of K+L. 2t + 3g+l 4 3 2 — -ooooo 04 1 + •00020 12 + •02002 74 1 — '10032 17 2 +•00835 45 3 • — '0002 2 44 4 — 'OOOOO 92 bum — '07197 26 •00068 29 2( + 3g-m. + •04217 86 • — -00068 + '00005 63 ooooo 01 + •00212 35 + 'OOOOO 01 0000T 65 + •02223 28 + '00001 65 0006S 38 + •02116 77 — •00068 38 OOOOI 56 —•00348 !5 — 'OOOOI 56 •00000 01 + •00007 + '00000 67 ■ — 'OOOOO 1 56 Mr. Ernest W. Brown, Theory of the Characteristic kY. Values of K+L. «'. 2i + g + m. 2i + g' -m. 4 + '00000 04 — 'OOOOO 19 3 + "OOOOI 83 — 'OOOIO 2 5 2 + "00090 10 — '00000 oS —•00457 80 + "OOOOO 40 I + •02712 OS — "00007 43 -•10787 ] 1 + "00030 28 O + •04871 79 — '00030 12 + •04461 77 + "00031 18 — I — "01092 69 + •00136 02 + •00213 57 — "00077 72 — 2 — -00329 91 + •00001 45 + 'OOT08 00 — "OOOOO 49 -3 — "OOOII 59 + •00000 02 + •00002 95 —4 — 'OOOOO, 26 + "OOOOO 08 Sum + •06241 36 + '00099 86 —•06468 98 — "00016 35 Values of i. (i 3 l)i. (/■■V;,:. (F/t'7,),-. {VJc'v')i 4 3 + 'OOOOO 04 — 'OOOOO 22 2 + 'OOOOO II + '00003 83 — '00020 05 1 + "OOOOO 74 + "00008 48 + '00343 98 — '01413 28 + •00180 29 + '00264 45 + •26780 3 — "26499 7 — 1 — "16631 44 + •10195 2 + •01283 48 + •01063 3 — 2 —•0084s 96 + -00444 82 - -00045 07 + •00015 08 -3 — -00005 95 + '00002 14 — "OCOOO 54 + "OOOOO 14 -4 — 'OOOOO °5 + '00000 01 — 'OOOOO 01 Sum —•17302 37 + ■10915 2 + •28366 — ■26854 7 Motion of the Moon. 5/ 123. Characteristic ke 2 a. Values of K+L. 21. 2 9 7 + 'OOOOO 10 S — - oooo5 78 3 — •00924 98 1 —•50561 36 — 1 +•33265 is 2! + g+2C. 5 -7 -9 Sum — -11496 98 + •00857 86 + •00130 57 + •00003 01 •28732 41 — 'OOOOO 95 —•00075 30 + ■00000 T4 + -ooooo 03 —•03408 84 + •00010 98 + •00003 19 — -14174 34 — '00052 08 + •00244 75 +•23592 66 + •00565 45 —•00237 97 + •08570 37 — •00163 11 + •00092 44 + ■02136 26 — -00008 48 — -00003 °5 + '00044 80 — '00000 10 — -ooooo 38 + •00000 48 + •00099 01 +•16685 14 + ■00352 80 — 9 Sum — -ooooo 02 — -00005 82 — •00553 48 — -26471 64 — •30003 95 — •26242 77 + •05329 6t + •01064 96 + •00023 8i + •00000 29 -•76859 OT + •00000 01 + •00001 32 + ■00101 68 — •00456 62 + ■00689 56 — -0002I 48 40 - — 'OOOOO Os + •00311 Royal Astrox. Soc, Vol. LIT. Mr. Ernest W. Brown, Theory of the Characteristic ke 2 a. Values of 2/. (ke- a )i. (AVV>*. (ka'a)i. 9 — 'OOOOO 02 7 — '00002 31 — 'OOOOO 10 S — 'OOOOO 10 — '00241 69 — '00016 04 3 — '00025 49 -•05586 70 — •01707 71 i — -03006 16 — •20534 29 — '09528 85 — i + •08743 57 + •02665 51 + •21904 30 -3 + •10025 96 + 'OOI4I 20 + •02244 01 -s + •00501 11 + •00001 32 + •00077 23 -7 + •00010 56 + "OOOOO 01 + 'OOOOO 74 -9 + 'OOOOO 10 + "OOOOO 01 Sum + •16249 55 -•23556 97 + •12973 59 124. Characteristic kerf a. Values of K+L. 21. 2i + g- r e + m. 2J- .-g — c — m. 9 + "OOOOO 4 7 + '00003 — "00006 1 5 + -00213 9 — "01062 8 + "00007 6 3 + '11381 3 — 'OOOOO 3 — '40554 7 + •00611 1 + 3-68451 4 + •00250 2 + ■03619 5 — •04405 2 — 1 — '01515 8 — -02278 5 + •00627 8 — '00129 8 -3 - -16887 4 — '00097 — -12646 4 + •00011 1 — 5 + -02886 5 — '00030 8 — •00363 7 + 'OOOOO 2 — 7 + -00074 8 — "OOOOO 3 — -00006 7 — 9 + -ooooi Sum +3-64608 7 — "02156 7 -•50392 7 —•03905 1 Motion of the Moon. 59 bum + - oooo i 9 -f -00038 6 — ■01159 3 —•29154 S + •01202 9 + •03803 6 — '02610 6 — '00076 4 — 'OOOOT 4 — •27955 2 Characteristic ket-'a . Values of K+ L. + •00000 2 + •00028 s + ■01677 6 —•01348 j + •00033 8 + •00002 9 + •00394 8 + •00000 7 + '00044 1. + '02328 2 + '74244 1 + •00857 • _ •06503 7 + '07677 2 + '00203 5 + •0000 1 8 + 78853 I — 'OOOOO 2 + -OO043 3 — '02096 I -■OI387 3 •03544 8 Values of zi. (7«?)a):. 7 5 + •00004 4 3 + •00452 5 1 + ■39617 9 — 1 -•08295 — 3 +•28615 6 ~ '5 + •00498 4 — 7 + •00003 9 hum - '60898 (*«y<o,\ (Z'€Tj f a) ( '0062 (/te'7ja);. + 'OOOOO 9 •0004 8 s + •00000 8 + '0009S 2 '°5375 8 — '00047 6 +•08783 6 •°33 1 7 -•03291 9 — -4212 •i33 2 + •1682 + •23485 •oi555 7 — '12480 + •01082 7 •00015 7 — ■00374 8 + '00009 6 -'OOOOO 2 — '00003 6 •0866 6o Mr. Ernest W. Brown, Theory of the 12=;. Characteristic ~ke' 2 a. Values of K+L. 21. 21 +g r 2m. 2i + g- -2m. 2i + g. 9 + '00000 3 + 'OOOOO I : 7 — 'OOOOI 6 + •00007 3 + '00009 3 5 — '00094 7 + ■00215 8 + 'O000I 2 + •00520 9 — '00000 1 3 — '°3544 6 + '00004 7 + •02615 5 + '00103 4 + •18452 3 + '00004 4 i , --28833 7 + •01564 3 + •20918 6 + •06701 3 -•10637 3 + •02244 8 — i + •14056 4 + 'o6S4S 3 — '22082 6 — "00171 1 — •09188 9 + ■00184 8 -3 + ■11342 9 -■00751 3 + •06191 8 + '00002 3 — '16626 7 + •00094 4 -5 + •00284 1 — '00005 ^ + •00239 3 — 'OOOOO 1 — '00906 2 + ■00001 1 -7 — 'OOOOO 3 + '00004 8 — '00019 9 -9 — 'OOOOO 4 1 Sum — ■06791 S + - o7357 4 + •08110 8 + ■06637 : -•18396 8 + •02529 4 I Values of 2i. (/.„»«), (l-v' 2 a) L 1 1 5 1 1 3 i 1 1 —1 1 1 -5 -7 — '00002 6 — •00215 4 — '07106 2 — •18423 6 + ■05465 + •00026 7 + 'OOOOO I + '00006 5 + •00201 2 + •11048 3 + •18843 7 + •02052 8 + •00017 3 + 'OOOOO 2 Sum — '20256 i + •32170 I (£f1'«)i. + '00000 2 + ■00014 9 + ■01187 9 — '02612 1 + •07789 9 — ■06604 4 —■00069 8 — 'ooooo 6 — '00294 o Motion of the Moon, 61 126. Characteristic Fa. Values of E+ L. 9 7 — -ooooo 01 5 4- - ooooo 65 3 4- '00040 11 i 4- '01996 76 — 1 4-74206 40 -3 -•i8S34 79 -5 4- '00258 76 -7 4-'oooio 36 -9 4- 'ooooo 16 Sum + •57978 40 4- -ooooo 05 4- -00005 98 4- -00810 84 — '00058 52 — '00003 09 — 'OOOOO 02 +•00755 24 2i + g. 4- 'OOOOO 01 4- -ooooo 88 4- -00077 21 + -05639 87 4-2-48539 91 4-1-66872 82 + -12295 52 4- -00276 64 4- -00004 10 4- -ooooo 07 ■•ooooo 07 • •00008 47 -•01078 67 •-0I733 5 1 -•00038 94 -•ooooo 23 +4-33707 03 -•02859 89 Values of 2i. (^)i. 1 7 ; ! 5 4- -ooooo 01 1 3 4-'ooooi 17 ! 1 4-'ooi32 75 — 1 + -i9 2 43 08 4- -16488 20 -5 4- -00249 9 1 _ 7 4- -oooo 1 21 Sum + •36116 33 (Wo),-. + •ooooo 02 + 'OOOO 2 57 + •OO424 96 + •77503 01 — [■41930 03 + •O4209 68 + ■00022 04 + •ooooo 14 — •59767 61 62 Mr. Ernest W . Ukoavn, Theory of the 127. ( lieaxteierktie kea 2 . Values of I{+ L. 4 + '00000 I 3 + '00013 8 2 + •00567 * I + ■14568 3 O "•03785 2 — I + •20850 9 — 2 -•°355 2 -3 — '00131 5 —4 — '00002 9 Sum + '28528 7 + '00007 6 + '00810 o + •00671 5 — •02037 7 + ■00131 5 — '00004 8 — '00421 9 + '00002 5 + '00105 4 + •03302 6 + •27137 4 — ■04866 7 -■07390 1 — '00479 ° — '00014 ° — 'ooooo 4 + •17797 7 + •00001 5 + •00152 3 — •03529 1 — '02091 6 — '00240 4 — '00026 =; -■05734 1 Values of (*«•').- + ■ooooo 2 + •00016 O + •00954 3 — ■00952 1 — •16281 9 — ■01633 8 — ■OOOI I -■17909 o + •00003 - + •00238 4 + •08233 9 + •05848 6 — '02730 .1 — '00036 7 — 'ooooo 4 + •11556 9 Motion of the 'Moon 128. Characteristic )s.e'a?. Values of K+L. i. 4 2i + g + m. 4- -ooooi 2f-rg — 111. — -ooooo I 3 — -00008 4 + •00053 9 4- -ooooo 4 2 -'01155 4- -00005 4 + •01512 i 4- -00048 8 1 -•84138 2 4- -01252 8 4- -04220 3 4- -05 264 —•00439 4 4- -031 15 9 — -00709 1 + ■01538 3 i — 1 + •02933 s + •05394 8 — •02860 8 -•00535 1 1 — 2 4- -00117 6 — -00267 6 + •03479 3 4- 'OOOOO I -3 — •00031 2 — '00002 4- -00058 ', ~~ 4 l Sum — 'OOOOI » 4- -ooooo 5 — '82722 4 + •09499 3 + •05755 5 + •06316 5 " Values of i. (.«?«'.);. 1 — 'OOOOO 2 2 — '00049 6 I -•09352 2 O 4--I2870 — I -•17592 4 — 2 — •00054 -3 — -ooooi 6 Sum — -14180 (A„'a-) r 4--ooooi 2 4- -00066 o 4-'oiio8 7 — •06371 + •18589 4- '00476 3 4--oooo2 8 + •-13963 Haverford College, L'a., U.S.A. 1900 May 14. (To be continued.) ■■:.* Theory of the Motion of the Moon ; containing a New Calculation of the Expressions for the Coordinates of the Moon in terms of the Time. By Ernest W. Brown, M.A., Sc.D., F.R.S. [Received 1905 January 21 ; read 1905 March 10.] PART IV. CHAPTERS VII. -IX. The previous parts of this memoir have been published in the Memoirs of the Royal Astronomical Society under the same title in 1897, 1899, 1900. The solution of the problem undertaken — the motion of the Moon as disturbed by the Sun supposed to move in a fixed elliptic orbit — is completed in the present part. It was stated in the introduction to Part I. that the main object in view was a new and accurate calculation of every coefficient in longitude, latitude, and parallax which is as great as one-hundredth of a second of arc, the result not to be in error by more than this amount. So far as I am able to see this plan has been carried out. A careful examination of the magnitude of the coefficients which would arise with characteristics higher than those calculated here, and a comparison with the results of Delaunay and Hansen, seem to show that no characteristics which would jriv ,'e coefficients so great as o //- oi have been omitted. There are possibly four or five terms whose characteristics are of the sixth order, which approach o'^oi quite closely. These omissions are, however, quite unimportant from a practical point of view. In a comparison between theory and observation a few such terms produce nothing sensible m the differences : it is only when the number of them is great that any effect is shown. The coefficients in longitude and latitude in every characteristic calculated have actually been found to o"-ooi, and in parallax to o"x>ooi. In the former case a large number of coefficients between o" - oi and o //- ooi are included; and, in fact, it is not difficult to see that there are comparatively few coefficients lying between o //- oo2 and o"-oi which are not included in the tables at the end of Chapter IX. Thus the theory, for purposes of comparison with observation, is considerably more accurate than was contemplated in the original plan. Moreover, a similar remark applies to the coefficients in parallax lying between o /; -ooo2 and o"*ooi ; thus the parallax of Royal Astron. Soc, Vol. LYII. i mini S2 Mr. Ernest W. Brown, Theory of the the Moon can certainly be found theoretically from the new table within o"'oi, so far as the solar perturbations are concerned. To complete the whole problem of the lunar motion, inequalities arising from other sources have still to be considered. These consist of the very minute terms arising from the parts of the solar disturbing function which are noted in Chap. I. § 4, the terms arising from the figure of the Earth, and perhaps from that of the Moon, the indirect planetary inequalities and the direct planetary inequalities. The kst is the only set which presents serious difficulties at the present time, and an investigation of them has already been started. Hill's work * on the inequalities produced by the figure of the Earth probably needs but little supplementing, while a new method f for investigating indirect planetary inequalities should render the task of calculating their coefficients comparatively easy. An important question which cannot be left aside is that of the accuracy of the computations by which the results have been obtained. It appears unlikely that the problem will be again completely solved in the near future, and some assurance is needed that the new coefficients, especially where they differ from those of Hansen or Delatjnay, are the correct ones. Fortunately the three theories are so entirely inde- pendent in their methods that agreement between them all amounts to practical certainty. The differences between Hansen and Delaltnay have given rise to much discussion in the past. In general my theory confirms the results of Delaunay in these cases ; the coefficients in which all three theories differ are those which are difficult to determine owing to the presence of very small divisors. The older methods, which approximate along powers of m, are theoretically less likely to be correct than a method which approximates along powers of the other parameters where the convergence is quite rapid. This, however, is a rather different question than that of the actual accuracy of the numerical work. For the latter very numerous tests have been used, covering almost every detail, as well as large masses of calculations. These tests are discussed below in Chap. VII. Sect, (iv), as well as in other papers to which reference will there be found. To return to the special part of the work now published. For the terms of the fifth order in u the homogeneous equations were used for the first time. In spite of the fact that this was a change of method involving much extra work the compu- tations were thus kept within reasonable bounds ; the expansion of Kii/r s would have nearly doubled the actual work done. The non-homogeneous equation was still used for z, as most of the multiplications had been obtained in calculating the fourth order terms. But for the sixth order in both u, z the non- homogeneous equations were used,, and the work was much less than had been expected. The final steps consisted in the transformation to polar coordinates, the change of * Washington Astronomical Papers, vol. iii. pt. 2, 1891. t E. W. Brown, Trans. Amer. Math. Soc, vol. vi. (1905). Motion of the Moon. 53 the arbitrary constants to the Delaunay system, and the insertion of their numerical values to reduce the results to seconds of arc. For the first of these the formulae \ were found to permit of such arrangements that much previous work could be usefully I utilised. In the final reduction to numbers, values for the constants were used which I were neither those of Hansen nor those of Delaunay. A selection was made from \ modern determinations which I believe will be found to be very close to the more I accurate values to be found when a thorough comparison of the completed theory with I the observations has been undertaken. I Owing mainly to the complicated character of the work which is embodied in the 1 results below I have been obliged to do much more actual calculation myself than % heretofore. All computations which could with advantage be turned over to a I computer have again been done by Mr. Ira I. Sterner, A.M.* His speed and I accuracy have been fully maintained, and have contributed in no small degree to an I earlier conclusion of the work than I had hoped. He has in all spent some three I thousand hours on these calculations, extended over seven and a half years ; my own I share I estimate at five or six thousand hours since the work was begun on a complete I plan in 1895. I The following is the table of contents for the whole memoir : — I Chapter I. — General Development of the Theory. I Section (i). An investigation of the disturbing function used, with the necessary I corrections. I Section (ii). The two forms of the equations of motion. I Section (iii). Development of the disturbing function according to powers of J 1 /a', z, e'. 1 Section (iv). The form of the solution. The general system of notation adopted I to represent the coefficients, arguments, &c. I Section (v). Method of solution. Preparation of the equations of motion. I Section (vi). Exact definitions of the arbitrary constants used in the theory. I Section (vii). Methods used for the solution of the equations of condition satisfied I by the coefficients. The long and short period terms which give rise to small divisors. 1 Manner of obtaining the higher parts of the motions of the perigee and node, I Section (viii). Details concerning the numerical calculations and the methods used i to verify them. I Section (ix). Transformation to polar coordinates. I Chapter II. — Terms of zero order. Numerical results. I Chapter III. — Numerical results for terms of the first order. I Chapter IV. — Numerical results for terms of the second order. % Chapter V. — Terms of the third order. I§ p * The expense has been met by grants from the Government Grant Committee of the Royal Society. 54 Mr. Ernest W. Brown, Theory of the Section (i). A brief outline of the application of the general method to terms of the third order in the calculation of the series A. Section (ii). New method for solving the linear equations when the series A have been obtained. Section (hi). Modification of the method in order to avoid, as far as possible, the loss of accuracy arising with long-period terms. Section (iv). The method of calculating the new parts of the motions of the perigee and node, and the coefficients arising therewith. Numerical values of certain quantities. Section (v). The final numerical results for the series A and for the coefficients of all terms of the third order in u, z. Chapter VI. — Terms of the fourth order. Section (i). Formulas and methods of procedure. Section (ii). Values of %, u^/sX; A, izjak. Chapter VII. — Terms of the fifth order. Section (i). Preparation of the equations for u, s. Section (ii). The new parts of c. Terms with small divisors. Section (iii). The equation for z. Section (iv). Nature of the computations. Tests for accuracy. Section (v). Values of A, B, w A £ -1 /ax ; A, izjak. Chapter VIII. — Terms of the sixth order. Section (i). Formulae and methods of procedure for u. Section (ii). The homogeneous equation for z. Section (iii). Values of A, B, w^ _1 /aA ; iz/&\. Chapter IX.— Results in polar coordinates. Section (i). Formula? for transformation. Section (ii). Change of the arbitrary constants. Section (iii). Numerical values of the constants. Section (iv). Numerical values of the parts of the arguments and coefficients arising from the various characteristics. Section (v). The final values of the coefficients in longitude, latitude, and parallax. Errata will be found at the ends of Chapters V., IN. Motion of the Moon. 55 CHAPTER VII. Terms or the Fifth Order. Section (i). Preparation of the Equations for u. s. 129. The Homogeneous Equations for u, s. As stated above, it was found necessary, in order to keep the calculations within reasonable limits, to change the method from the non-homogeneous to the homogeneous form of the equations, as far as u r „ s 5 were concerned. Equations (6), (7) of Chap. I. are those to be used. They may be written — • o = <J>EEf+^m 2 L' + ff = D-(us + z-)-Du . ])s—(Dzy-- 2m(iiDs-sDu) + }m 2 (u + s) 2 - 5 mV + 3 u 2 + 4u 3 -D- 1 (D'w 2 +I)'co a ) (i\ o = •* = f + ijmlP-' A' +1)- > S(M' ) = nDs-sDu~2WMs + D- l 'im i (u i -^)-^-^ + ',^"J r ^^ cm <)u (2) where the constants of integration are omitted, since they contain only characteristics of even order, and q takes only the values 2, 3, since characteristics containing a 2 are neglected. The new symbols must be defined ; in all cases, unless stated otherwise, suffixes represent the orders of the characteristics present in the functions to which they are attached. The majority of the indices of £ contain c, g. Xow, to the order considered here, C = C + C 2 + C 4 , g = g + g 2 +g.„ and therefore the operations J), I) 2 , D~ l introduce parts of c, g other than c,„ g ; these parts must be separated. Put f = D\us + ~J)-Du . Ds-(Dz) 2 -2m(uDs-sDu) + %m i (ii + s) i -3mh\ F = uDs—sI)u—2mus+%-m i D~ i (ii?—s' i ), L'= 4 - {3co., + 4w ,-i)-i(Z<' w , + i)' w ,)}, 9m- " ' 3m 2 ( ' 'dn du os os j ' where c a , g are substituted for c, g in the coefficients when the operations D, I> 2 , D~ l are performed. Hence Sf, S(Df') are respectively the parts due toe — c , g — g in D ll {vs + ^)-J)v, . Ds-(Dz) 2 —2m(u.Ds-sDii)—D- l {D'to. 2 + D'ui i ), D(v Ds—sDh — 2mns). Denote the unknown coefficients of £ ± (- ,+Tl in it^/n by A.,. X'_ .,-. The equations $ = o, F=o are linear with respect to all these coefficients, since terms of order higher :fi Mr. Ekxf.st W. Bkowx. Theory of the than 5 are neglected and those of lower order have been found. It is necessary to put the equationslnto forms convenient for calculation; the method is implicitly contained in § 33. It will be seen from that section that if we equate to zero the coefficients of f i+T in 2(1 +m) I [<t> + (1 + 2 m)¥]=o, * D^—i — 2m-Hnr ±> (3) the terms of principal importance, involving the unknowns in the left-hand members, are a(X, + X'_ ( ) ; -a{\-\'^), respectively. This fact serves as a guide for the arrangement of the equations which now follows. 130. Let 9m = D{uf l ) = T 1 (Du-u), Ss = Z>(sif) = £(Ds+s), P = D 2 — 1 — 2m + y, (21 + 7)-— 1 — •?»' + so that at*,), ^'o arc divisible by m 2 and then it will be found that IrnH +|mV+s2 + L')+.V(i + 2 m)D-i| M 2 -s 2 + A' + ^ 2 S( J Df')| (4) ^ = w S s _ s5u _ 2 (i+m)«a+^!i>- 1 |« 2 -^ + A'+^(^')} ... (5) It is to be noticed that all terms except those in a> 3 are homogeneous products of the second order with respect to u, s, and their derivatives. There are terms whose characteristics are of orders o, 1 , 2, 3, 4, 5 in "» *. -■ Let u a) s a , s a (a = 5, 4. 3) and «!» s *<> s * ( 6 — S-«) distinguish the orders, it being noted that c =o and therefore that z, is not present. Then 2 S /n*/\1 and we easily obtain from equation (5), 2>-i* = -£<**-«*> +*-' [{IC^-^^-^^+^+f^- 1 i^ + A' + ^f)} (6) Eauations (0 are immediately derivable from (4), (6). 1. 1 Form* for Computation.-^ the following formula* the bar over any expres- .ion, at usual, means that i/< has been put for £. The sign 2, denoting summation for values of a, is omitted for the sake of brevity when no misunderstanding of the meaning can arise. awL Motion of the Moon. 57 mi. ill i II m P mm P P 1 1 ill Put G = M a % + M <( % + -W H = M a M 6 _7{^r 6 4..i.A' + -^/(-Of) ... .A. -7> (8) A=- .-'/ . ■--«■ — .-/( -zDz a . Dz b + 2z A + (2m-im*)2z A + M+'im*-G + i ) m%- S + 2m)D~m (9) B = u a $s h +u a &s„-f 2(1 +m)|-F -1 A+ 2s (1 ^ + m. 2s B S;, + 3in 2 (i + 2m)Z) _1 H-^2(i + 2m) ... (10) Then the values of X„ X'_j are obtained by equating to zero the coefficients of £ 2 ' +r in the equations » a * J -Z)- : J5 + .^ + Jli , -'-A = o (11) ^F 6 + Z)-' 1 B+r a » 6 +J-F^A = o (12) It will then be found that the principal coefficients of \ h X'_ t derived from the left-hand member of (1 1) are respectively a, o, and from (12) o, a, while all the other unknowns have coefficients small compared with a. 132. Method for Approximation. — Denote the difference between the iih and (■i — i)th approximation to any function Q by Q w . The first approximations to G-j H, A, B are obtained by neglecting all the unknowns in these four functions. For G-, II have m s as a factor, while u ;> . %u s occur elsewhere in A, 15 only when multiplied by 9--s , which also has m 2 as a factor. In (n), (12) all the unknowns except those we are considering are multiplied by terms which also contain the same factor. Hence G (1) , H (1) , A (1) , B (1) , \p\ X'_i (1) are obtained from the known terms in the equations of § 131. The second approximation is obtained by substituting u 6 a) for u b in the various formula?. Then « 6 (S) is determined from : « 6 '% — «5 a, «0> G (2) = M s ll) M + tt 5 ,1, tt 0> H« : •3w 5 a) . 5« -"^ d) 7Sr +-|m J G (a) + 3m 2 (i + 2 m)i)- 1 H®, '. B <2) = u 5 m Ss + u s ™$8 + 2(1+ m) AF- 1 A' 3) + 3ni 2 ( 1 + 2m)D- 1 H e >-f- 2 (i + 2m), « 5 «» . ar 1 + M5 a) (so-aC" 1 )-^" 1 B <2) +AI , - I A tS ' = o, lT^.&i- i +i^(s -a,i- 1 ) + D- l B' S) +l'F- l A^ = o. The further approximations proceed in a similar manner. Exceptions to this method will only occur when the divisors introduced by F" 1 , I)- 1 are such as to render it useless, or very tedious, owing to the number of approximations required. For such cases a special method is devised in § 138, below. 133. Development ofV, A'. — These functions depend solely on the terms in co 2 , co h where (§8) o> 2 = |m 2 (« 2 a 2 +s 2 fl 2 )+im 2 Mi'6 2 — m 2 s 2 6 2 , a'w 3 = |m 2 (M 3 a 3 + s 3 a 3 ) + fm 2 (M 2 sc 3 + Ms a c 3 ) — sm 2 s a (MC 3 + sc 3 ). iHence 1/ = w 2 a 2 + s 2 a 2 + I«*6 2 -|« 2 62 + ^" 1 {-^{uW& i +s?Da 2 )-%u / 8m i + i& i Db l ) + *? {¥(w s a3 + s 3 a 3 ) + f(M 2 sc 3 +tts 2 c 3 )- ■ ;:s 2 V«^ + «3j, -D^i-MuWuz+^Mz) + 1 (ifoDc 3 + mWlj) - 1 s*(nDt 3 + sDl,)} A' = it?a 2 -s%+ -T {f(« 8 a3-s 3 a a )+;J(tt%C3-$ 2 MC 3 )-s 2 (w 3 -sc s ) (A '• v S^^w^^w^^I^^w^^^pwS^^^^^^^^^^^^^^^v™ 5 8 Mr. Ernest W. I'.eowx. Theory of the The terms in these expressions are all known, for the German letters having suffix 2 contain ef as a factor, and those having the suffix 3 have a as a factor. These are expanded (§ 9) in powers of e', while M s =(« , ) 4 +(tt , ),+ (t« , ),+ (u%+ (m j ) , &c. The terms whose characteristics are of order 5 are chosen out to obtain I/, A'. 1 "4. Development of Sf, 8(Z>f ). — The calculation of these functions is troublesome and a more detailed exposition is advisable. We have (§ 129) Sf = c„ oc dg oc 2 de cf 3 where the partial derivatives refer only to the coefficients in f, and c , g are put for c, g. The terms containing c 2 g 2 , g/, g 4 are absent, since f t does not contain g. The new ]>arts of c contained in c 4 are as yet unknown ; they will be determined in Section (ns. and meanwhile it may be noted that they are only present in the coefficients of £'' il - The expression for Sf arises (§ 129) from Sf = 8{D%us)—Bu . Ss-(i + 2m)(u$s-s$u) +Q} , wllGTG Q^B-^-iD^-B-^B'^ + D'o,,) (13) To find 3f 3 /3c put «„r l = 2/tj4", ih£ l = ^,J' h , where a = 3, 2 ; h = 3-0 ; q-q t = zi + r = p ; and where the summation signs refer to all the terms present. Then PO 9f 3 = 2?'£ [{p i + qqi+(i + 2m)(q + q l )}i l ^ qi + {p , + 2(?,-(i + 2m)( 9 + g,)}^- 3 / J -J + -g^ 3c dc 1 ■H-iH-J + OC the summation sign referring to all possible values of p, q, q x . Since p = q-qi, the first factor may be written •* do 3c \3c e'e y Also S^^ + fi^,) = S(Si,M ±M6S.«)- Omitting the summation sign for the two values of a, as in § 131, and putting (0=3, 2; U = 3-a. E i:' = M^^ttA jy =S45« ±S^W, a the results can be symbolically expressed in the form in which the sign 5' denotes that in the formation of E the proper coefficient arising from d p /dc-dq°/dc is to be attached to the corresponding term in E. and similarly for ■ mm. Motion of the Moon. 59 the other two terms. The above algebraical expression appears more complicated than it is in actual calculation, for the derivatives of p, q, q 1 are integers less than 4. By a quite similar investigation and with the same notation we obtain I- (Di z >) = - 22' f(i + m) f E + i?' DW + l? Pi . dc L dc dc 00 J The derivatives with respect to g are obtained in replacing d/'dc by d/cg in the above formula?. The same formula? serve for oi\/d c , o(D^)/dc, but they simplify. We have Qi = o. a=i, b = o, q =p — 2t±c, q 1 = o; the only terms present are those of characteristic e. Finally, in a similar manner, we obtain 3 2 f, 2(M S + S e U ), -W-^- = — 2(M e S -S e M ). d. 135. Development of 3Q 3 /dc, aQ s /dg.— Differentiating (13) in the previous §, we obtain from the only possible combination of a, b, which, in connection with § 133, gives the required formula. For the derivative with respect to g of the first two terms of Q we have Now z only contains £ in the combination £ 2 <+'-_£-2<-t i Let j' denote the value of ^ when this expression is replaced by £" +T +£~ 2i ~ T . Then, since only the first multiple of g occurs in z 2 , z u d Hence c'g L eg 'J dg Section (ii). J"Ae New Parts of c. Terms -with Small Divisors. 136. Determination of c^ — In those terms of the fifth order containing £ 2i±c a new part. c it of c arises for determination, and one of the unknown coefficients is indeter- i , ire minate ; the definition of the latter (§ 25) is such that the coefficients of £ ±c in « 6 £~ to be equal. One of the unknowns is thus replaced by c 4 , but the linear character of the equations is retained. The formulae for finding c 4 and the coefficient of £ c must, Royal Asteon. Soc, Vol. LVII. k 60 Mr. Ernest W. Brown, Theory of the however, be quite differently formed from those for the other unknowns. The main reason for this is that in the case of £ ±0 we have since c is put for c after the operation F" 1 (§ 130) ; the divisor is therefore very small. If all powers of m had been included in F the divisor would have been zero and approximation impossible. Hence the formula; must be so arranged that F _1 is not present. Equations (10), (1 1), (12) of § 131 may be written B = B' + (i+m)F 1 A, B = iD2(M a s,-sX), A =-F(ms + « 2 ), giving A-— — F2)2(M a s 6 -* a w 1) ) + ^£_=o ... ... ... ... (14) 2 + 2m i+m - us -z i +~ - — D2{u a s„-s a u b ) = o ... ... ... (15) i + m 2 + 2m equations which are free from the operator F _1 , since A, B' do not contain it. These two quantities contain c 4 . To isolate it put (see equations (9), (10)) A = A' + c 4 i^».+(r + 2m)2)->|-(2« 1 ')} 1 [_ v C C ) Substituting in (14), (15) we obtain C4 r^k + I ! + 2 m + -?_ iD^hm,')] = -A' + .-l {^2(«A~ ««« t )-B"} ... (16) [_dc I 2 + 2m) cic J i+m o = -j«.-s 2 + -i-rB"-i2>2(M A -« a « 6 ) + ic 4 2>- l9 -(i« 1 ')] (17) i+m L cc -■ 137. These are well adapted for solution by continued approximation. Equating the coefficients of £ c to zero in each of them, it is to be noticed that the unknowns in A', B", $(u a s b s a u b ) have the factor m 2 at least (§ 132). If such terms be neglected in the first approximation, (16) is a simple equation to find c 4 . When c 4 has been obtained Ao + Xo', =2X , is found from (17), since ~a(X + X / ) is the cofficient of £>, while the other unknowns have the factor nr. When c 4 , X have been obtained the remaining \„ X'_, are found by the ordinary method given in § 132. The first approximation completed, the resulting values are used for n 5 , s s to obtain a second approximation to c 4 from (16) and to X from (17), and then to the other unknowns from § 132, and so on. There is less disturbance of the computation sheets than would appear. The first approximation to A is found, exactly as in § 132, by omitting all unknowns. In that Motion of the .Moon, 6 1 to B wo omit all unknowns and the term 2(1 + m)-'-F- 1 A in tlio coefficient of £ c , and then all the quantities for (16), (17) are to hand. A difference occurs in the second approximation, due to the fact that ■-S' " ( Z)f '')> 3m 2 do as 9*i must be respectively included in H (2) , A (2) (§ 131) for the unknowns other than 138. Small Divi-wrst. — The operators 7) -1 , F" 1 introduce small divisors in the cases of long-period and monthly terms, respectively ; each set of terms whose argu- ments differ by 21 involves one of the former or two of the latter. As the basis of the continued approximation was the ability to neglect terms having m- as a factor, approximation may become impossible, or is at least very slow, when the small divisor is of order m 2 . The method used for this case is the same as that of § 29. Taking the case of a long-period term, suppose that the corresponding coefficients arc X , V- The first approximation to the other A,., X'_ f is first obtained with X , X,/ considered as unknowns, so that they are expressed as linear functions of X , X ' and a known part. The first approximation to X , X ' is then obtained by using these values of X i; \'_ { in w s , a« 5 , &c, instead of neglecting them. The process thus leads to two new simultaneous linear equations for X , X '. When these are solved a substitution gives the first approximation to the other unknowns. In the application of this rule it was found sufficient to determine X ±1 , X' Tls in terms of X , X ', a second approximation giving the required accuracy, and the other coefficients being determined by the ordinary method. For the two monthly terms, corresponding to coefficients X () , XJ, \_ u X/, the same process was followed with respect to that pair of them, say X . X,/, which had a divisor containing m 2 as a factor : the divisor of the other pair has then only m as a factor. Section (iii.) The Equation for z. 139. The non-homogeneous form was retained for s, since the greater part of the series-multiplications and many of the additions had already been obtained in com- puting the terms of lower orders. The method for z is, therefore, the same as in previous chapters. The known part of the expansion of Kzj-fi (equations (16), (18), § 20) is expressed in the form .^^"B.+^'JJ. + ^'B.+^'B^ with *," = -U-.S, r Po Then, using the notation, ^(,7' 7> ~)=f( u : 8 > *)-r-( M o. »o> Po). 6 2 Mr. Ernest W. Brown, Theory of the for brevity of expression, we have B 1 =[u l +s 1 ]-~(u , s ), JB i= [u 2 + S 2 — f(« 1 a +8 l 2 )-|« l S l +» 1 !! ]-T-(Mo,So ) Po)» B 3 =[U 3 +S 3 -|(M 1 M 2 + S 1 S 2 )-f(iM2 + M2Sl)+2«l% + M(V + S 1 3 )+¥(«^ 2 «l+% S l 2 )-| ;:; l ! ( M l+ S l)J -r( M 0! s o> Po)> ^=[^ + 84 — f(2M 1 W 3 +M 2 2 +2SlS 3 +*3 2 ) — I( W i«3 + M 3*l+ M 2 S 2) + 2SiS 3 + »./ + 3 /(Mi 2 M2+Sl J S 2 )+VX M l 2s 2+ M 2Sl 2 +2 M l S l M 2+2«hSlS2) — |»l 2 (M 2 + *2) — 5«l ;S 2(Ml+*i) Tlie terms duo to 12, are givc-n by u) 8) -A = m h {b 2 + :; * 7 («c s + «,)} , in which b 2 . c 3 , c, are expanded in powers of </ and respectively multiplied by those terms in 2, uz, sz which give characteristics of the fifth order. Finally, the part of -D 2 %z^ which depends on c-c , g-g is -2S'(c 2 ^+ g|?) Dz 3 - S fri - 2g,^l', with the notation for 5/ given in § 134, and for z 1 given in § 135. All the terms are known with the exception of g 4 ; this quantity is determined as in §§31, 73. ■ m till Section (iv). Nature of the Computations. Tests for Accuracy. 140. The Computations for u M s K . — The equations, prepared for computation, have been given in § 131. The principal part of the labour is the formation of the products u a u b , u a s„, 5u ti . Bs b , u a $s h , Dz a . Dz h , z a z b , (a = 4, 3 ; 6 = 5 — a) in which s b =u^ £,?,_,— — iW For calculating I/, A' (§ 133) we require the products "o M M U a %. «a »i» u a % ««) U<x u b «a where a + /^4 in the first three and a + b + c<4 in the last two. Most of these, however, are deducible by briefer multiplications from the results obtained in calcu- lating the fourth order terms for which the functions Xu a u b /u \ Zu a $ b /u v s 0l &c, were obtained ; the multiplications of these by u 2 , ti % &c., arc easy on account of the rapid convergence of all functions of it , s . For computing Sf, S(J>f) (§ 134) we require u a s b , s„ $u„ s 2 « u »i Dz 2 , w, s , s Bui ; (0=3, 2 ; 6=3— a) the first, third, and fifth of these had been previously obtained. The products in § 135 which contain an accented z are deducible by mere changes of sign from those formed without an accent. Of the remaining calculations which do not consist of additions or subtractions there are multiplications of series by constant factors {e.g. G by -Jm 2 in A). The operation D consists in multiplying each coefficient by the corresponding index of £, is § m m M, •/!'•/} of /A, .IA./.?7. 63 while the operations J9 : , J)~ 2 , F 1 consist of similar divisions. In many cases the factors are small and the divisors greater than unity, while the number of significant figures is small, so that these operations can frequently be done without the use of logarithms. It is, in fact, much easier to divide mentally by a number of. sav, three figures than to multiply by such a number. Crelle's or Tamborrel's tables might have been used for both operations. As a matter of fact these aids were not employed. I believe that Avhere the numbers to be multiplied consist of only three significant figures, and do not run at all consecutively, it takes less time to use a four-place table of logarithms (which can be mounted on a single card) than to be obliged to turn over the pages of a bulky volume of multiplication tables. 141. The Computations for .:. — The processes were the same as those of previous chapters. Of the k"„ 7?* all but k" ,, B. had been previously found ; K". r is obtained from the brief series multiplications of z t by -^i</2p s , while all the products and most of the sums in ll t had been obtained in computing u,,. The only multiplications remaining were the k",72 5 _, (i=i, 2, 3, 4). and the uz in tS^:, most of which latter products were at hand. The long-period terms do not produce small divisions, and the initial number of places of decimals adopted was such that special methods for the monthly terms were not required. The approximations also were sufficiently rapid. 142. Tests. — All multiplications and additions were tested by the addition test explained in Chap. I., Sect. viii. Practically every operation comes under one of these heads. The apparent exceptions are the operations I), D~ l , F -1 . For J) we have I'-r-Z (22t / i ( + r3,,. t ),'* f If we put £ = 1 we obtain a test easy to compute, since i is a small integer. The same process performed on the result tests for Z)~ ] and a slight extension enables us to test similarly for F -1 . This method works well for the details of the calculations. The calculations en manse can be verified by going back to the differential equations and using the above method for the operator D with £ = 1. This was not used very extensively on the fifth-order terms ; for reasons which will appear in the rest of this section. Such a test is very laborious and it seemed hardly necessary to make it. 143. An indirect test in fact arises with every set of terms with a given characteristic. Each such set is u, s, and every other set in z has one (long-period) or two (monthly) terms whose coefficients possess small divisors, and the process of division by these is practically the last step in the determination of the coefficients. Xow, in the fifth-order terms the chief danger of error arises not in the detail* of the calculations, which can be tested by the method just described, but in the possible omission of a whole set of terms, or in the use of a wrong set, owing to the very large number of sets to be dealt with. It is just such errors that would be detected, for even though they might not be very large the numbers which go to make up the long- period or monthly terms are always formed of the differences of numbers large in 6 4 Mr. Ernest W. Brown, Theory of the comparison with their algebraical sum, so that an error, after the process of division by a small divisor, may cause a coefficient to appear several times its actual value ; a rouo-h comparison with Delaunay's or Hansen's results would reveal the error at once. It must be remembered that such errors would practically run through the whole series of calculations, but would be mainly revealed by a few terms. Thus one of the chief causes of the extent of the calculations furnishes the most valuable test of their accuracy. Even if Delaunay's results were not available the test would often still work, owing to the fact that negative powers of m cannot be present, and therefore that the maximum order of magnitude of any coefficient can be roughly stated in advance.* 144. A Special Set of Tests. — In a paper f published in 1896 I gave some extensions of Adams's theorems which connected the mean motions of the perigee and node, not only with the constant term of the parallax but with one another. In fact, equations were obtained relating the parts of c, characteristics k 2 , e 2 k 2 , k 4 , e' 2 k 2 respectively, with those of £r. characteristics e 2 , e 4 , e 2 k 2 , e¥ 2 . The former set is determined in connection with u from the equations for u, s (§136), the latter set from the non-homogeneous equation for z. There were thus four separate tests from these relations. Further, the coefficients of certain characteristics in the constant term in the expression for the parallax are related by separate equations to parts of the motions of the perigee and node ; these characteristics in the parallax are e 2 , k 2 , e 4 , eV 2 , kV 2 , k 4 , e 2 k 2 , furnishing seven other tests which, however, with the exception of the first two, are not very searching, owing to the small number of places of decimals used in the results for the fourth-order characteristics of the parallax. The relations in question are included in the equations do, , a [E+mv> _ I 1 - 2 de 2 \ (18) (i9) where ft, /3 3 , /3 3 (c 1; c 2 , c 3 in my previous papers) is the set of canonical constants complementary to the constants giving the position of the mean Moon, its perigee and its node at time t=o ; tt u B x are the mean motions of the perigee and node ; and (Q)° represents the mean value of a function Q. 145. I have already developed $ a method for the calculation of &, /3 3 solely from the coefficients in u, s, ft-2 The following are the results ■1 1844 44e 2 — -02324e 4 — -26363e 2 k 2 — -oonoeV 2 , na '■\— —2-00205 9k 2 ~i-96376k 1 — -28546e 2 k 2 — -oo568e' 2 k 2 . * An error, the use of u^~ l instead of w 4 /« in B 4 , was actually first revealed in this way, although the difference between these two series is divisible by m 2 . f " On Certain Properties of the Mean Motions," <fec, Proc. Lond. Math. Soc. vol. 28. X " On the Formation of the Derivatives of the Lunar Coordinates with Eespect to the Elements," Trans. Amer, Math. Soc. vol. 4 (1903), pp. 234-248. Motion of the Moon. 65 Also \ 1 +111/' V i+m/ The values of c , g„ are given in Chapter III., those of c 3 , g 2 in Chapter V., those of c 4 , g 4 in the following Section. They are c= + 1-07158 32774 + -00268 57ie 2 — -03465 6oe' 2 + -o538s 595k 2 --o22i2 6a 2 + , ooo23e 1 + -oi8ic 2 e' s + , ooi45e 2 k 2 + -i77oe' , k ! + -07657k 4 , g= +ro85i7 i4266 + -oo3i8 6i83e 2 + 'oos64 6535e' 2 - -00806 6255k 2 + -omo 58a 8 + -ooo2 7e 4 + -o 1 04e V 2 + 008 7 5e s k 2 — -oo9oe' 2 k 2 — -00883k 4 . 146. Differentiate (18) with respect to k 2 , and (19) with respect to e 2 ; the two left-hand members should then be equal. Calculate the terms with characteristics 1, e 8 , k 2 , c' 2 in these members ; the results, which should be the same, give the following differences : (/3j) e *e k a = — -oo6-?7 804} ■>.„, V-^/e K 4 ^'f I (Jiff. = -00000 002 J (As)k=ge= = —-00637 892) (Pi) "k»c.. + (/3 3 ) e = ka g e= + 2(/? 3 ) k =g 1! . = --00291 1 diff< _ , ooooi . 2(/3 , 2) a <o ka + (/3 2 ) «c . ka = — •00290J (/3s)e»k»Ck» + (/3 3 )e=k»g k » + 2(/3 !! ) e ,C lt , = - "03252 | ^g. __ . QQQ0I , 2 (A))k«ge' + (fia)vg*°V = ~ -03251 ) W^ + (C^«- = — °»8}diff. = . O002 • (PsWge'-KfrlVgeV = — -0226) here (/3 2 ) e . denotes the coeflicient of e 2 in fi-,/nd 2 , &c. These differences should properly be divided by 2, because (fi 3 ) k >, which is accu- rately determined, is very nearly equal to 2. The first result tests certain terms of the third order ; the other three results test terms of the fifth order. The latter involve a large proportion of the forms for computation of the fifth-order terms and of the results for the terms of the fourth and lower orders. 147. Next, if we equate to zero the coefficients of 1, e 2 , k 2 , t' 2 in (18), (19), the coefficients of 1. e' 2 in the constant term of i/r should be zero, and, since E+ M = n 2 a z . i^K = ~(i +m)n« 3 Q° e . 1 '(&),, ,g k . = -(1 +m)wa 3 fy\, j From the results of the transformation to polars (Chapter IX. ) I find / a \° I I = +"99999 97i + 'ooooo oe 2 + *ooi2 e' 2 + 'ooooo k 2 + -ooa 2 + oooi e 4 — -oooi eV 2 + •0040 e 2 k 2 + -ooo e' 2 k 2 — -0049 k 4 . It will be noticed that the differences from zero of the coefficients of e 2 . k 2 , eY 2 , t' 2 k 2 do not exceed one unit in the last places calculated. A corresponding agreement will be found for the coefficients of e\ e 2 k 2 . k 4 . when calculated from (20). This decree of accuracy should be expected, since the number of places of decimals used in trans - (ao) ^fBi 66 Mr. Erxkst AY. Brow>~, Theory of the *'*°**llSl forming to parallax is smaller than that actually obtained in u, z. These tests apply to the results for the fourth and lower orders.* Section (v). Values of A, B, u£ l /a\ ; A, i~ A /a\. 148. The following tables show the characteristics and arguments of the terms of the fifth order which have been calculated, together with the §§ in which the results are given. I have not set forth the types of coefficients, since these are sufficiently evident from the arguments and characteristics according to the scheme adopted in Section (iv), Chapter L, and illustrated in the later chapters. § A Arguments. 149 e 5 2t±5C, 2t±3C, 2i±c 150 eV 2i±4C±mj 2-i±2c±m, 2i±m 151 eV* 2i±30±2m, 2t±3C, 2i±c±2m, 2i±c 152 eV 3 2*+ 20+301, 21+20+111, 2i±3m, 2i±m iS3 e 3 k s 2l±3C±2g, 2i±30, 2t±C±2g, 2*±C iS4 eVk 3 2i±2C±m±2g, 2i±2C±m, 2t±m±2g, 2i±m 155 ee'% 3 2i±C±2Hl±2g, 2»+C+2m, 2l±C+2g, 21 + C 156 e' 3 k 2 2i±3m±2g, 21+301, 2i+m+2g, 2i±m iS7 ek* 2i±c+:4g, 2i+e+2g, 2i+e 158 e'k> 2t±m±4g, 2i±m±2g, 2i±m 159 e 4 <x 2*^40, 2i!±2C, 2% x 160 e 3 e'a 2"ii±3C±m, at^+c+m 161 e 2 k 2 a 2t' l ±2C±2g, 2t 1 ±2C, 2li±2g, 21, 162 ee'k 2 a 2i,±o±m±2g, 2t,±o±m 163 kV 2-i]±4g> 2*l±2g» 2*1 I 164 : k 5 2i±Sg, 2i±3g, 2%±g J 165 k 3 e 2 [ 2 i±3g±20, 2i±3g, 2i±g±2C, 2*±g 1 166 k 3 ee' 2t+3g+c+in, 2i+g+e+m 167 k 3 g /2 2i±3g±2m, 2i±3g, 2i+g+2m, 2»+g 168 ke 4 2t±g±4C, 2i±g±2C, 2t±g 169 keV 2i±g±3C±m, 2i±g±c±m 170 keV 2 2 *+g±2c+2m, 2»+g+2c, 2i+g±2rn, 2i+g 171 kee' 3 2i±g±c±3m, 2i±g±c±m 172 k 3 ea 2t,±3g±:c, 2i,±g±C 173 kVa 2*i±3g ±m . 2i!±g±m 174 ke 3 a 2t 1 ±g±3C, 2ii±g±C J 175 keVa 2i 1 ±g±20±m, 2i 1 ±g±m * On the subject of this section see two papers by the writer : " On the Degree of Accuracy of the New Lunar Theory," &c, Monthly Notices, April 1904 ; " On the Completion of the Solution of the Main Problem in the New Lunar Theory," ib., December 1904. Motion <>f tin .Vflon. The terms with the following arguments and characteristics required the method of § 138 to obtain the approximations with sufficient rapidity : — Arguments. ;( 2 — c + 2m), :(c-2g), =(2 + C+2H1 — 2g), ~(z — e + 2m — 2g), -1, :(i-c + m), :(i + 2C— 2g), Characteristics. eV 2 , ee' 2 k 2 e 3 k 2 , ee' 2 k 2 , ek 4 ee' ! k 2 ee'% 2 e ,4 «, e 2 k'-«, k 4 n eVa, ee'k 2 a e 2 k 2 a ee'k 2 a. ±(i+c+m — 2g), For other terms where the approximations were slow it was found to be sufficient to calculate the third approximation, since a regular law of decrease then appeared which permitted the remainders to be written down from inspection. The results for A/a 2 \, B/a 2 \. u^jak are given in the following section under the columns headed A, B. u£~ l /ak. The last is a change of plan from previous chapters, intended to increase the clearness of the reading, since the results are always referred to their arguments and not to the special notations adopted for the coefficients. The choice of subsidiary results for publication at the present time was less easy with the homogeneous equations than with the ncn -homogeneous. It may be stated, in this connection, that long-period small divisors do not appear in A or B except in terms which have the explicit factor m 2 . while the monthly small divisors occur in B but not in A. Thus certain coefficients in the B-tablcs are not accurate to the last figure set down, but it is better to retain the same number of places for every coefficient under a given characteristic up to the final results. The work easily shows the extent to which the latter are correct, and in the tables for u^/ak none are given beyond this point. It is to be noted that A = A, B = --B, so that, given the coefficients in these functions for £ 2i+T , it is not necessary to write down those for £ Si ~ T . The coefficients of g in B are enclosed in square brackets, to signify that they are those parts of B denoted by B" in § 136. In Chap. VI. the zL-tables for the ^-equation were given in two parts (see § 98) ; here the parts are not separated, since the Z A arc generally quite small and easy to compute, and, further, they were not separated on the computation sheets. All the other arrangements of the tables are the same as those of Chap. VI. Royal Astron. Soc, Vol. LVIL 68 Mr. Eenest W. Brown, Theory of the 149. Characteristic e 5 . Values of A B i. 21 + 5c. 2i+3C. 2« + C. ' i. 2S+5C. 2J+3C. zi + c. 4 + -00003 4 3 + •00007 + ■00094 3 — - OOOOI — '00014 + •00007 + •00246 + •01912 2 — 'OOOOI — •00035 — '00486 1 + •00256 + '05054 — -OOII2 1 — -00036 — '01208 + •00015 + •05446 —•00888 + - OOOI9 — •01258 + ■01396 [+•01315] — 1 — -02304 — -00029 + '0002I — i + •00824 —•00093 + •00443 — 2 + -00038 — •00007 + -OOI24 — 2 — '00089 + •00051 — '00222 — -00003 + ■00047 + •00371 -3 + ■00001 — '00054 — '00102 — 4 + •00004 + •00039 + -OOOI9 — 4 — 'OOO04 — -oooii — -00003 Sum... + •03444 + •04469 + ■02451 Sum... —•00563 +•00045 + "00946 '-f-ae" 21+ 5C. 2J-SC. 21+ 3C. 4 ! + •0001 + •0002 + •0001 + '0008 2 + -0004 — 'O002 + •0041 — •0004 1 + •0004 — -0004 + •01 01 —■0054 — '0020 + •0103 + •0005 — -0124 + •0010 — '0042 — •0042 — 1 — -0096 + •0030 + •0005 + •0036 + '0006 — 2 + •0018 + '0001 + •0002 + ■0002 — 3 + -ooot —4 Sum... + •0029 + •0001 + •0012 + •0016 — •0015 — •0050 c . = +-00023. ■ mi 1 m m m I IS 1 i Motion of the Moon, 69 1 50. Characteristic eV. Yiiliu. s of A i. 2i + 40 + m. 21 + 40— m. 21 + 2e + m. 24 + 20 — m. 2s + m. 4 + ■00001 — "00004 3 + '00002 — '00018 + ■00091 — •OOI79 2 — *oooi8 + -00099 — •00591 + •02674 — •02614 I -•00734 + •02727 — •08748 + •23138 — '03696 — •10854 + •12325 -•°7559 + •00218 — '02908 — I — •04499 — -02582 + •00422 — •02991 + •01844 — 2 — •00227 + 'oo 1 66 + •00869 — -00282 + •10569 ~3 + •00130 + •00025 + ■01943 — '00340 + •00957 -4 + ■00134 — •00016 + •00151 —■00025 + •00029 11 Sum... -•16068 + •12746 -•i353i + ■22484 + •03998 B «'. 2s + 4c + m, 21 + 40 — m. 2% + 20 + m. 2J + 2e-m. 2i + m. .■: '■' 4 3 + 'OOOOI — '00007 + "00027 2 + '00003 — •CCC.Gij + •00091 — •00433 + •00859 I I + '00112 — '00420 + •02694 -•08567 + -00043 ! 1 O + •03275 -•04343 +•00945 + •02907 + •00054 ;;li;l;vll:lll — I + ■01128 + - OI442 -•00563 + •00055 + •02974 1 — 2 — •00225 — •OOOI5 —■00336 + -00I2I — •04085 j 111 — 3 — '00119 + •01032 — '00766 + -O0I30 — ■00158 1 111* -4 — •00050 + "00008 — '00027 + 'OOO03 — '00003 ! 1111 Sum... + •04124 -■033 5 + •02039 -•0579I — •00289 1 M e v£ -1 -r a eV i. j I 2i + 4a + m. f 2i— 4c— m. 22 + 40-- a. 2t — 4c + m. < t + '0004 — -oooi ; * 1 + '6016 + -0003 1111: - + '0007 + 0001 + - ooo6 i — •0010 — "0016 + '0032 — 'ooog c ) ■ — •0258 — •0016 + •0322 + •0021 — I — •0287 — •0001 — 0216 + -0002 ~~ '. /: - : + •0103 + 0013 ' : ' ; :> ; -* : //: + •OQ03 + 0001 -4 : ; ;::::::■ Sum — •0449 — - ooo6 + • OI 53 + •0022 mmMmmmlmSm^ 7° Mr. Ernest ^Y. Brown, Theory of the ..^-'-r-aeV (continued) i. 2s + 2c + m. 2»— 20-m. 2*' + 20— m. 2i — 2e + m. zi + m. 2i— m. 4 + •0002 — •0001 + •0001 i 3 + •0060 + •0001 — ■0010 — '0003 + •0014 i 2 —•0009 + -0045 + -0036 — •0014 —•0072 + •0307 1 1 — ■0224 —•0005 + •0626 —•024s — •0207 — •0062 1 — •0472 —•0051 —0138 + - ooo8 + -0405 — '0298 1 — 1 + •0193 —•0014 + •0294 + -0044 + •0084 — '0060 1 — 2 + •0024 —•0005 + •0001 + •0022 — "0005 ' ! -3 -4 + -0005 — •0001 + •0001 Sum... -•0483 + •0037 +•0813 —•0217 + •0230 -•0103 1 151. Characteristic e'V 2 . Values of i. 21 + 30 + 2m. 2S + 3c — 2m. 2J+3C 2t + c + 2m. 2( + C-2m. 2J + C. 4 1 + •0001 3 + •0001 +•0055 — •0019 2 1 + •0002 + •0068 — •0027 —•0025 +•1347 -•0514 1 + ■0030 +•1315 -■0631 -•1787 +•5806 — •2882 1 — '0484 +TS23 -•1234 -•0179 —•0163 + •0170 -1 -•0537 -•0035 —•0079 -•0638 —•0038 + - 0235 — •0271 + '0048 — •0041 +•2655 -•0637 -•0939 -3 + '0345 — •0063 —•0078 + -0407 —•0013 -•0130 -4 + '0038 — "O00I — •0012 + •0014 — -0004 Sum... -•0877 +•3156 — '2I02 B +•0447 +•6358 -•4083 i *■ 2i + 30 + 2m. 24 + 30— 2m. 2J + 30. 2* + c + 2m. 2i + o — 2m. 2i + c. ! 4 1 3 ; :^0£§% — •0005 + -O002 i 2 — •0007 + '0002 — •0016 -•0181 + •0074 1 1 — '0013 -•0205 + •0100 -•0337 — •0616 + •0859 — •02IO — •0282 + •0540 + •0058 -■0887 [ + 'IS9S] — 1. -'003 1 -•0235 + ■0361 — •0189 — •0424 -'0473 ! — 2 + ■0120 — •0039 + •0026 -•0599 — •0123 + •0308 -3 — 'OO9O — ■0014 + •0033 -•0053 — •0005 + •0017 i -4 — - 0004 — •0001 — -0002 | Sum... — -0227 -•0783 + •1062 -•II38 — '2241 + •2382 Mutton of the Moon. 7i •i_ .., : ^-aeV I. 3 24 + 3c + 2m. 22 — 3c — 2m. 24 + 30 -2m. 21 — 30 + 2m. 2i+3e. 21-30. + '0l6 — •002 — -004 2 1 + •033 + •001 — •021 + •014 i — ■013 + •016 + ■015 — '007 —•020 — - 0I2 — '002 • + •075 + -008 — -042 ?':\ :; ^*^S' :/•••. — 1 -•113 + ■078 + "001 — •046 — j + •029 — "004 — "010 -3 + •002 Sum... -•094 + •034 + ■166 + '001 —•105 -•013 :::■■ t. 2i+e + 2m. 2J — c— 2m. 28 + C— 2m. 2*-0 + 2m. 2» + e. 2J-C. 3 + •006 + ■001 — •002 2 — ■001 +•118 + •018 — •020 — •007 -•O38 1 -■051 + •042 +•235 — '04 T — - I0I -•068 — •063 -•035 + '027 + •091 — -xoo — 'I0O — 1 + •072 — ■007 -•094 + •028 + •080 — •010 — 2 + ■014 -•003 + •001 — "004 — •001 -3 Sum... — •029 + •124 +■184 +•059 —■132 — -219 c e v = + 'oi8i. 152. Characteristic eV 3 . Values of i. 21 + 2C + 3m. 22+2C — 3m. 24 + 20 + m. 2J+20— m. 2«'+ 3m. 2s + m. 4 3 + - ooo8 —•0003 + •0001 2 + •0289 + •0015 — ■0148 + '0064 -•0193 1 + '0010 + •4412 +•0039 -•2390 + '0396 + •0623 -•0588 +•1698 —•4221 + ■2731 + -0994 + - 399 2 — 1 + ■6702 -•0565 -•1867 +•4332 + 7555 —•4246 — 2 + •2282 + •0134 —•0876 -•0586 + '4i73 -•1979 -3 | + •0813 + •0014 -•0330 -•0057 + '0192 — •0088 -4 j + •0031 — '0012 + "0005 — 'O002 Sum... + ■9250 + •5990 —•7252 + '3 8 79 + I- 3379 — •1892 i 72 Mr. Ernest W. Brown, Theory of the B i. 2 i + 2c + 3m. 2* + 20 — 3m. 24 + 2C + m. 28 + 20 — m. 28+ 3m. 2t + m, 4 3 — •0001 2 — •0017 — -0004 + •0013 + "0004 — •0030 I — - ooo4 + •0242 — •0069 + •0096 + "0046 + •0178 O — "0291 + •0075 -'"45 +•1379 + •0278 -•0457 — I + •043! + -0086 -•0156 -■0348 +•1365 -•0423 — 2 + ■1104 + •0038 — •0287 — '0223 — -0023 + •0072 -3 — - OOII + •0002 + •0010 — •0010 — - OOII + •0006 —4 — •0002 + •0001 Sum... + •1227 + •0426 -•1650 + •0907 +•1659 -■0655 «w» 'C~ l — aeV i. 28+20+ 3m. 28 — 20 — 3m. 24 + 20 — 3m. 2J — 20 + 3m. 2J + 2C + m. 21 — 20 — m. 3 + '012 -•005 2 + •260 + •003 + - 007 — -091 1 + •138 + •059 — •OI7 + •064 — "021 — •005 + - I43 + ■045 — •162 -•°4S — 1 -•537 + ■071 + - 004 + ■029 — 2 i 3 + •065 -•030 Sum... -"493 + '4°5 + •276 + 039 -•163 -•077 i. 2J+20 — m. 24—20 + 111. 2i + 3m. 24— 3m. 24 + m. 2% — m. 3 — 'OOI + •002 — 'OOI 2 — 'OOI — •022 + •001 + ■058 — - 002 — -026 1 -•028 + •323 + - 02I + •678 + '021 — "269 +•135 + •071 — '224 + ■215 -1-045 +•782 — 1 — - 6io — 'OOI + "220 + •003 — '060 + •023 — 2 —•003 + -O03 — 'OOI -3 Sum... -•507 + ■370 + - 02I + •956 —1-087 + •509 Motion of the. Moon. 153. Characteristic : e 3 k 2 . Values of 73, i. 21 + 30 4- 2g. 2J + 30-2g. 21 -i- 3c. 2;-.- r + 2-j;. 2i'+C-2g. 2« + C. ' 4 — "OOOOI — 'OOOOI 3 — 'OOOOI — "00002 + "00004 — "00022 — '00015 2 + '0OOO2 — "OOIIO — •OOOO7 + "00024 -•OI654 — -00488 1 + '00099 -•05196 4- "oo 1 7 7 -■01714 + '00407 +•04333 + •00943 + •00377 + '42905 -•839SI + ■00270 — '00006 — 1 -•09563 — '00030 — "00666 + •02984 — •28820 + ■00854 -> + '00264 — "02607 + •00274 + •00175 — •OO762 —02586 -3 — '00015 — •00078 — •00387 — "00096 — •0O0O2 — '00207 -4 + •00011 + "OOOOI — ■00021 + "00006 + '000OI -■3OS83 — '00003 Sum... 1 —'08259 — •07644 + •42273 —•82598 + •01881 , 1 B !. i 2*'+3e + 2g. 2J + 3C — 2g. 28 + 3c. zi + e + 2g. 28+C — 2g. 2t + e. 4 ' — 'OOOOI 3 — "OOOOI — '00002 — '0004 1 — "OOOIO 2 | — -00005 — "00086 — •00014 — •00185 — •01289 -■00579 ' 1 —'00297 — "02605 — '00869 — '07990 + '02099 — -11843 —'12680 1 — •01865 — ■24098 +•04525 — '02231 [ + •24621] — 1 ~-o<>,^9 — '01191 + "04258 + ■01589 + •01792 + •02627 : — 2 ' —"00665 + "00234 — •OO183 — "00009 — •01684 — •02208 i -—3 — 'OOOII — 'OOI2I — •OOI76 — "002 20 — '00039 — •OOI31 — I — 'COO I J — "OOOOI — 'OOOII — "OOOo6 Sum... — '07330 —•05636 -•21093 — "02298 -■01394 + •12477 j We*- t 1 ~ - ae 3 k 2 i. 4 • 2i + 3e + 2g. 2t-3C-2g. 21 + 3C — 2g. 28-30+2g. 21 + 3C. 2J-3C. + •0001 — "OOOI j 3 i + '0003 — "OO02 — '0020 2 + "OOOI — *00O2 — "0096 — "0064 I 1 — •0212 — "0164 + "0204 + "O00I + "0003 + '0005 + '0414 -■0332 + "0478 + "08l2 + ■0652 — 1 —•0292 + •0008 + •0034 — "OOIO — •0035 + •0030 — 2 + •0104 + - ooo6 + '00OI -3 -4 — '0003 + "0004 + "0001 is;: ^o-;y;i- ;.|^ ;'.'•'• 1' Sum... -•0186 + •0215 —•0454 + •0574 + •0780 + '0600 1 74 Mr. Ernest W. Brown, Theory of the u eV £ _1 +ae 3 k 2 (continued) 2J + C+2g. 2i — C-2g, 21 + C — 2g. 2t — C + 2g. 2J + C. zi — e. 3 — ■0003 — •0001 — •0005 2 — '0029 — •OO57 — •0017 — •001 1 — -0151 I — '004I + •0040 — - oo6i -•0939 — -onS -•0280 -•2427 — '0019 + •0612 + '2074 — •0946 — '0946 — I + -i3 r 9 + - 0253 — •0025 — •OO98 + •0570 4-'0224 — 2 -•0053 + •0005 + ■0052 -•O005 + •0019 + •0017 -3 + •0001 + -0003 Sum... — "1202 + •0247 + •0521 + •1015 -•0483 — '1141 c oV = + 00145, 154. Characteristic : eVk 2 . Values of A f'. 21 + 2C + m + 2g. 2S + 2C + m — 2g. 21 + 20 — m + 2g. 24 + 2C — m — 2g. 4 + •00001 : 3 — •00002 + '00001 — '00005 • 2 i — '00003 + •00092 + '00015 — •01053 1 — '00069 + •05303 + '00339 — •16500 + ■01234 +■38403 + •02137 -■54035 — 1 — •22251 -■25505 — •00741 + ■01095 — 2 -•06377 —•03567 + '00224 + •00588 ~3 + •00236 — •00032 — '00138 + •00006 — 4 + •00055 + "00003 — ■00016 Sum ... -•2717s + •14695 + •01821 — •69903 «. 2t+ 2C + m. 2J + 2C — m. 2J + ra + 2g. 2!+m — 2g. 28 + m. 1 ! 4 — '00003 + '00002 1 3 + •00003 — "OOOOI — 'OO0O2 — •00056 + '00030 2 + •00061 — '00238 +'00019 + •00092 + '01034 1 1 + •02468 -■02687 +"03053 + •15664 — '10067 ! +•42785 — •47278 —-01030 -•14557 -•33o55 1 - 1 + •10576 —•44694 —'81041 — -14192 — -19920 i — 2 — -07460 — •02770 —-03083 — '00061 — ■06463 ! -3 -•01376 + ■00113 —'00003 + ■00007 — •OO203 ; —4 — '00034 + •00003 +'00007 — •00007 Sum ... + •47023 -•97552 --82080 — '13106 -73649 ■ HP mm mm Motion of the Mo»n. U 75 i. 2i t- 2C + m + 2». 2S + 2C + m — 2g. 2S + 2C — m+2g. 21 + 2C — m-2g. 4 3 + '00003 — •00016 ! 2 + '00005 + •00234 — •00037 — -00707 I + -00285 + •05964 — •01526 — '11226 !■■ o + •06395 + "00685 -•07843 +•02514 1 . — I' + •04286 +•05958 — •02791 + '03346 — 2 —■00288 -•03142 + ■00113 + ■00389 -3 -•00551 — - ooio8 + ■00078 + '00015 —4 — ■00029 + '00004 Sum + ■10103 +•09594 — ■12002 -•05685 >*. 4 3 2 1 o — 1 — 2 -3 -4 Sum 2i + 2C + m. + 'Q000I + '00096 + ■05111 + •31953 — •OI784 -■ 6593 -•OO757 — '0OOI5 + •28012 ji-jc-- in. — '0000 1 — '00282 — •12626 -•31454 + ■01876 — -00962 + •00186 + '00002 -•4j26l 2j + m + 2g. + '00001 + '00132 + ■03039 — ■ 10006 + •04470 — ■06061 — •00299 — 'OOO06 -■08730 2i + m-~2g. + •00078 + •01530 + '00096 + '161133 ■-■16277 -•00835 --'COOK) + •01509 21 + m, + -00039 + ■01883 — ■01694 + •02876 — •20480 -•05815 — 'C0I2iS -•23319 u eW . T'+acVk 2 p" «. 22 f 20 + m + 2g. 2t — 2C-m — 2g. 2i + 2c + m-2g. 2i- •ac— m + 2g. 2l + 2C-m + 2g, 28 — 2e + m — 2g, 4 + '0001 j|lli!lilfil?l:?; 3 + •0035 — •0001 -•0015 ^^^S9mfi 2 + "0043 + '0001 -•0055 — 'OOOI \ 1 — '0840 + '0258 -•1856 + '0002 — •0077 Wgggg:; + '0001 — '0489 + ■1474 — ■2308 + '0011 + •0676 ^S0llMfxfMx>ii. — 1 —•0816 — ■0013 — •0327 — ■0014 — '0008 + ■0057 mjm^M^M — 2 + '0500 + •0229 — '0003 — ■0049 + '0001 pill -3 + '0007 + •0005 — '0003 -4 + '0001 Sum ... —•0307 — •1263 + ■1640 -•4237 — '0047 + '0641 Hoy At AsTitox. Soc, Vol. LV1I. M r 7 6 Mr. Ernest W. Brown, 27teo^ 0/ tf<e w aVkJ ^-f-aeVk' (continued) i. 2J + 2C — m — 2g. 28 — 2C + m+2g. 28 + 2C + H1. 2i— 20— m. 24 + 20 - m. 2i — 2e + m. ■ 4 3 — •0027 .:• .' : : •: . ' : ;i 4- '0002 ! 2 -■0015 4- -0007 + •0001 -■0851 -—•0004 — - OI26 i 1 — •1024 + '0154 + "0060 — •0627 — - oii4 -•3298 1 — -1113 + •2490 4- -2517 -•045S -■3S04 4- -0044 1 — 1 + ■0015 — •0190 — ■0154 ■—•0102 + •6405 4--o2i6 j — 2 — ■0027 + -0005 4--oo68 — '0002 —•0052 4- -0005 1 -3 — ■0001 4-"0022 —•0005 -4 :■' : ■-"■■■ ■' y y: : :S- : :i "■ ' y - : 4* 'OOOI Sum ... — ■2165 + •2466 +•2515 — •2064 + •2726 -•3157 »'. 2i-j-m + 2g. 2j— m — 2g. 2i + m— 2g. 2i— m + 2g. 21 + m. 2i — m. 4 3 — •0001 4- - oooi — '0003 i 2 + - OOOI — -0240 — ■0018 — 'OO03 4- -0019 — •0127 1. + ■0032 -•0536 — -0170 -•0174 — •0014 -■2568 + •0312 — -0490 -•0137 -■1218 4-7122 -•5019 — 1 + ■4615 — ■0236 4--H96 — -0690 4-'xo27 -•OSSO —2 -■0093 — - ooo6 + ■0034 4- -0008 4- "0142 — -0044 1 -3 —4 + •0003 — '0001 . 4- '0003. — -oooi 1 1 Sum ... + ■4870 -■1508 4- -0904 — •2078 4- "8300 -■8312 155. Characteristic ee' 2 k 2 . Values of A »'. 2i + c + 2m + 2g. 2i + c + 2m— 2g. 2i + c-2m + 2g. 2i + c - 2m - 2g. 3 4- "oooi 4- "0005 2 — •0001 — -0008 4- -0005 -■0179 1 — ■O0O2 + ■0033 4- -0105 4--I473 4- -0390 — •0123 + •0582 -•1332 — 1 — •3226 — -2290 4- 'ono — •oooi : ^- :-^~ 21- -•0546 — •0024 4- -0030 — -0009 -3 4- - oi44 4- "0003 — •oooi Sum -•3241 — •2409 4- -0832 —■0043 ■ Motion of the Moon. 77 o — I — 2 -3 Sum , 31 - (" -r 2IM. — "0007 — '0202 + •0625 + ■2677 — ■2448 — ■O078 + •0567 21 + c — 2m. — "0003 — -0181 -■3772 + 7/>=6 — -0326 + "coSo — ■0002 -■0598 21 ■ ■'. - 22. — - 0004 + "O003 + -OI37 + '0364 + •0312 — - oo8o + •0732 21 ■•■ c - 2C- — ■OCC.} T- '0046 -'°35 2 + ■0671 + •0867 + ■0018 + -IIS4 2t +C. + '0002 + ■0078 + •2580 + I464 + •1292 + - OSOS + 'CC2I + "5942 2f + + 2m + 22. — "0004 — ■0165 -■3784 —•0444 -•0052 ■■4449 B 22 + C + 2m — 2g. — '0009 —•0817 -•0493 + •0744 — "0089 — '0001 -•0665 2t + c — 2m + 2g. 2t + C — 2m — 2g. — -0007 — 'OOIO — '0204 — "0272 -■6451 + '1033 — •2671 + 1-3216 — •0042 + "0107 — '0002 — '0003 + 1-4071 -"9377 ' — , _J :/ : : :i,:: 2t + C+2m. 28 + — 2m. 2! + e + 2g. 2» + C — 2g. 2t + C. 3 + "0002 2 — '0002 - # oi37 + -OI2S + •0077 1 + -OI72 - -4509 + •0081 + -3700 + •2878 + •2883 — 1-2136 — -0228 + I -4800 [-•070S] — 1 -'9542 — -8189 -•0058 — 'O284 — '4794 — 2 -•l8 5 + 'o°53 + -0070 + "0020 + •0924 -3 -•OO39 — 'OOOl + •0029 + -00I9 Sum ... -•8378 —2-4919 — -0106 + 1-8363 — -1601 7 8 Mr. Ernest W. Brown, Theory of the Mo.-'=k=r' I -r-aee' 2 k 2 i. 2t + C+2IU+2g. 2! — c — 2m — 2g 2! + C + 2m- -2g- 21--C — 2m + 2g- 2J + 0-2m + 2g. 21 c + 2m — 2g. 3 + '002 2 + ■059 — - 00I + ■014 I — - 242 -•057 —•030 + 'OOI -•32 O •-•037 -•031 + '036 + •004 +•087 — I — '240 + '333 + '038 + •27 + '022 — 2 + •157 + •010 — •090 -3 + '002 Sum ... — •081 — •2l8 + •255 + •043 +•185 -•197 2J + — m 2g. 2i-c + m + 2g. 21 + + 2m. 2i — c — 2m. 2* - c + 2m. 3 -> — '002 1 —•361 — '001 — 'OOI + •070 — '°49 + • 1 6 1 — 1 —•098 + i'z49 + 1 "006 — 2 — •001 + '007 + '072 — 3 + "ooi Sum ... -•392 + I - 206 + 1*239 •043 •776 •5J5 •010 -1 '345 •002 •073 ■898 •892 •008 -i'S73 + "003 + "34i + i'433 + -n6 + -003 + 1-896 Sum 21 + C+ 2$ 21 — C— 22. 21 + C. I — I + •086 — 2 — •038 -3 — 'OOI + •047 + •034 -i'i76 •401 — - 002 — 'o6i — 'OOI + "OOI + •008 + •098 + -167 4- '009 + "038 — ■164 + ■006 -1-280 + •421 + '199 + •199 —-'007 — -060 — •827 + ro 3 8 — •071 ~ -002 •—•004 - -030 — -ooi — "OOI + 1-245 — -029 <W,= + -i77o. k Motion of the Moon. 79 156. Characteristic e /3 k 2 . Values of A ■ . siSiM 2i + 3m + 2g. 2t + 3m-2g. 2i + 3m. 2i + m + 2g. 21 J- m — 2g. 2t + m. 3 + '0004 — 'OOOI 2 + •0018 + '0002 + "0002 + •0025 — •0027 1 + ■0004 + '0349 — - oo8o + •0017 — •O436 — '0190 + •1256 + ■1251 — '2031 + •0174 + •0292 +•1769 — 1 + •3169 + •0186 - -9697 — -1714 •0000 + •6411 — 2 + •0920 + •0011 - '0504 — •0850 — "0004 +•0253 -3 + •0047 + •0001 — '00 1 1 — '0029 + -0005 urn... + -S396 + ■1816 — I'232I — •2400 — '0119 + •8220 B 1. zi + 3m + 2g. 2J + 3m-ag. Zi + 3m. 2S + m+2g. 2i + m— 2g, zi + m. 3 2 : ; : : / :V-::: ; : : \;:: ■ — ■0021 — - 000I + •0001 — ■0037 — •0013 1 — "0002 — •0187 — '0184 — '0003 + '0201 — "0116 i — •0698 + •1567 + -1648 -•1380 + ■1626 —1038 j — 1 -•2423 + •0103 -i - 5943 + •1286 -•OO57 + ■8841 ; —2 — - 02l6 —•0006 — -0320 + •0319 + ■0210 ! : -3 -—■OOO8 — "0004 + - ooo8 + "0002 Sum... -'3347 + ■1456 — 1-4804 + ■0231 + T733 + 7886 | v vv i~ 1 ^ ■ae 3 k 3 I. 21 + 3m + 2g. 2s-3m-2g. 2t + 3m-2g. 2i-3m + 2g. 24+ 3m. 2J-3m. 3 + •001 2 + ■021 — "006 1 1 + •258 + •036 + '001 — •203 + '002 + •02 7 — MI 7 + •009 + '204 —388 1 — 1 -•637 + •054 — 'OIQ + 1-289 + •003 ~*2 3 + - 034 + •001 + '002 + - 0I0 Sum... — - 60I + •307 — -026 — •007 + i'5°3 -•594 1 1 Mr. Ernest W. Brown, Theory of the n w^sj-^'-j-ae^k 2 (continued) i. 2i + m + 2g. 2i— m — 2g. 2t + m — 2g. 2i— m + 2g. 2i + m. 2t — m. 3 2 — 'Ol6 + •003 ] j I -■380 — •016 — ■002 + •097 ' | O + •110 -■150 — -001 + •091 + •014 j ! — I — 2 + •586 -•031 — •018 + ■027 + ■003 — •688 — ■006 + •003 | -3 Sum... + •555 -•286 -•184 + •029 — '605 + ""7 5 Sum.. + •00405 157. Characteristic ek 4 . Values of A i. 2t + C + 4g. 2i + c-4g. 2i + C + 2g. 21 + C— 2g. 2« + e. 1 3 — - oooo8 + - OOOOI — '00003 — •00002 2 + •00093 + -00005 + -OOI3I + •00007 1 + ■00002 + ■02617 + •00443 — •02682 — -01225 — '00024 - -04835 + - 03463 — •26771 + •05994 — 1 + •00379 — -00499 -•OOO95 + '0I4l6 + •02155 — 2 + •00061 — '00007 + - 003I3 — •OOO33 + •00218 ; -3 — ■00013 — •00075 — •00002 + •00013 — '02639 + •04055 —•27944 + ■07160 B i. 2% + c + 4S. 2i + c-4g. 2t + + 2g. 21 + C — 2g. 21 + C. j 3 — '00010 — 'OOOOI 2 + •00177 — 'OOOOI — -oooi I — -00058 1 — '00004 + '09268 — -00427 + -18160 - -04036 — '00213 + •00898 — •80476 -3 -I 5 6 94 [-1-04073] -1 + ■04491 — •00200 + ■36829 — -16987 + '29027 — 2 — ■00518 — 'O00O2 + "00192 — '00136 — '00736 -3 — •00017 — "00036 — 'OOOOI — '00008 Sum... + '°3739 + -I0I3I -•43919 —3-14669 - 79885 Motion of the Moon. tW^-T-aek* Si i. 21 + C + 4g; 2i-c-4g. 24 + C — 4g. 2J — C + 4g. 3 + '0002 2 4- "0009 — ■0028 I — -0604 + '0207 — 'OOOI O + •0014 -'4579 — •0027 — I — '0024 +•0959 — 2 + - 0082 — •0077 i -3 — '0002 ^^Pil|; : Sum + "0080 -■OS79 -•4424 + -0854 i. ai+e + 2g 2t~C — 2g. 2t + C — 2g. 2%— e + 2g. 2j + e. 2% — e. 3 • • !'— 'OQo'i. •'■ ^^Blii% ' 2 —•0074 — "00 1 1 + '0004 llilS^"lll|K$$ftK "$;■;'' I [ -•0034 — '2171 — '0074 +'0004 -•0805 O + •0013 +■48x3 + 4-S387 — 1-9841 H--3342 + ■3342 — I + '0032 + '0002 — -0484 + -7232 + 'ioo8 — -0109 — 2 -3 —•0089 + •0001 -■0043 — '0008 — -0017 — -0008 — 'OOOI ||p||l|fi!!tl>§x'l Sum... +•4705 + 4-2713 — 1-2700 +-4346 + ■2431 c u . = +'07657 .: : ■ < :i v : :' i:: . . : : ' : ; ; : : ' ; ■■>-[ 158. Characteristic e'k 4 . Values of A i. 3 2i + m+4g. 2t + m— 4g. 2s + m + 2g. 2t + m — 2g. 28 + m. | — •00015 — -OOOOI + -00006 — •00002 2 + •00730 + -00006 + •00526 — •00076 ■ ■ ■ 1. + •00002 — "0I2I2 + •00133 —•04200 — -00520 — '00025 + -00I55 + •03301 + •00136 — •05614 — 1 + ■04376 + "00005 + "02084 — •00414 - -05545 —2 — ■0081 1 — -01413 — -00015 + •00265 -3 + "00034 — ■00039 + '00003 Sum... + •03576 — •OO337 + •04071 — ■03961 -•11489 82 Mr. Ernest W. Brown, Theory of the B »'. 2s + m + 4g. 2j' + m-4g. 24 + m + 2g. 2t + m — 2g. 2i + m. 3 + ■00019 + •00002 — -ooooi 2 — -00464 + '00008 + •00316 —•00139 I + •05167 + •00825 + •00155 -•09713 O — •OOI57 + •00266 + "45! 95 -•50959 — •03209 — I — -09909 — -00005 + •08329 -•03351 + •22527 — 2 + •00821 — '01241 — ■00034 + •00230 -3 — •00042 — •00013 — -ooooi + •00001 Sum... — •09287 + •04983 + •53103 -•53872 + •09696 w A .f" 1 -T-ae'k* «'. 2t + m + 4g. 2i — m — 4g. 2j + m-4g. 2i-m + 4g. 3 + •0002 — •0001 2 + "0030 + •0013 I — -1301 + •0771 + •0020 + •0026 — I + "0007 + •0001 — - ooo6 — 2 + •0198 —•0154 -3 + -0003 —•0002 Sum + •0208 — •1249 + •0810 — -0162 t. 24 + m+2g. 2t— m-2g. 2! + m~2g. 2t— m + 2g. 21' + m. 2i— m. 3 — -0001 2 — -0040 + •0016 — '0001 +•0003 1 — •1026 + ■1115 — •0002 — ■0044 + •0036 + •0011 +•1848 -•1833 — •0024 —•1037 +•0735 — 1 + •0472 + •0022 -•0068 -•0650 +•1338 -•0387 — 2 +•0133 — •0001 — -0064 + '0002 -3 + •0001 Sum... + •0617 + "0803 -•0771 -•0740 + •0258 +•0387 Motion of the Moon. 159, Characteristic e*a. Values of 83 21. 21 + 4c. 21 + 20. zi. 1 1 7 + '0001 + 'OOOI ! 5 + "OOOI + •0013 -•0314 3 + '0022 - '°794 -•1150 i 1 — •0670 -•0931 + •0027 — r + •'415 + •0106 -3 + - 004I + -0098 1 — s + •0025 — •0320 ! -7 — 'OO29 -•0054 1 Sum | + "0805 -•1881 —■1436 J 21. 7 2% + 4c. 2J + 20. 2i. + •0001 1 5 + 'OOOI + '°°45 3 + "O002 + •0138 — 'OOOI 1 + -OIS2 — 'OIIO + •0238 \ — 1 — •O277 — - oi7i 3 + "OO08 + '0002 — 5 — - OOI2 + '0004 7 + '0001 + '0004 Sum — •0126 -■013? + •0283 'V.f''+ae 4 a / 5 3 1 -1 6 "S -7 Sum 24 + 40. — ■012 + •051 + •008 -•003 + '044 28 — 4c, — '001 — '0Q4 + '002 + -003 '000 21 + 20. — •OI3 — ■O23 + •087 — ■008 — '002 + "04I 28 — 20, 2t. ••. — 'OO I ■ : i — •OI3 — '005 > — •OI3 — "041 + ■036 + '020 ''.;| -•005 -■095 — "001 — •006 + •003 -■127 1 60. Characteristic eh' a. Values of 'A ■ ■ zi. ■'■■■■.■"■■■ ! 28 + 30 + m. 21 + 30— m. 2i + c+.m. 2t + c — m. •7 .+ '0009 .5 1 — 'ooor + '0015 + "OIIO + '0170 ' '3 '; | ' +'oi6i + -0275 + 'SSoz — "4322 I + 7904 - -1771 + 2-5540 - '1684 — I 1 --0381 -i"3735 + 557° + i'2i47 _ 3 I -'2193 — '5492 — -2981; + '3211 -s — ■0708 — -0196 - -1847 + -1151 — 7 — ■0196 + '0083 + -0034 + '0029 Sum + •4586 — 2'082I + 3"i936 . +I'07M Royal Astkox. Soc, Vol. LVII. 84 Mr. Ernest W. Brown, Theory of the Sum 2!. 2i + 3c + m. 1 ' 5 3 -'0059 i — •2621 — i + •1203 _ 3 -•0134 1 — 5 — •0220 — 7 + •0013 -•1818 2i + 3e — m. + '0034 + ■0198 -'3479 + -0400 — ■0180 — '0022 -•3049 — ■0031 -•I368 + '8794 — •0729 — ■O23O + ■0124 + '0006 + •6566 + ■0029 + '034I — •O464 — '1214 + •1002 -■0288 — •0007 — '060I « eV „£ -1 -*- ae3e ' a 2i. 2t + 3c + m. 2t — 3c — m. 2S + 3C-M. 28 — 3c + m. 2s + c + m. 21 — e-m. 28 + c — m. 21 — c + m. 7 — •004 + •002 — 'OOI + "OOI s -■065 — •on + -003 — •032 — '002 + -027 3 + •005 -■056 — •001 — -116 + -130 -•238 - - o73 + -179 1 + •183 + •003 -■039 — •190 + 1-064 — '106 - -171 + i'95 — 1 -•055 + '0I3 — •668 — •004 - -286 + •374 — 2-41 - -030 — 3 + •150 + '457 — •001 — "072 + •009 + -033 — -004 -5 — •014 — '004 — '002 + '002 — 'OOI -7 — •001 Sum + •268 — •109 -•255 -•320 + -837 + '006 — 2'62I + 2T22 161. Characteristic eVa. Values of A 21. ; 2t+2C + 2g. 2t + 20-2g. 21 + 2C. 2 i + 2g. 28. 7 5 + •0001 — •0043 — '0007 + •0026 3 — •OOOI — •02 72 + '0271 -•0243 +•5215 1 + •0304 + -0033 + 1-0340 + •4413 -■1873 — 1 + '3747 4- -0179 - -1647 +•0615 -3 + -0S40 + •1558 - '1047 -•0183 -5 + •0111 -•0058 + -08l3 — •0200 -7 — '0040 — '0002 — -0006 — - ooii: Sum + •4661 + 'I395 + -8724 + '4384 + •3369 Mutivn of flu- Mam. S.s I 21. j 21 + 20 + 2g. 24 + 20 — 2g. 2 i + 2C. 2» + 2g. 21. 7 1 + •0001 1 1 S ! 4- 'O002 + '0003 + •0067 i l 3 + '0013 + '0212 fOI38 + •0435 + •2328 1 I + ■0956 + •0894 + -40IO -^SSo + •;-'>$ — I i + '04IO -•07IS — •5640 + ■2047 I ~3 + ■0415 — •O32O + ■1863 + •1826 1 ~5 ! + "004c; + -0O49 + ■0339 + -0079 7 . + "0006 — 'OOOI + '0005 Sura... i + •1849 + '0121 + •0715 + '3°40 -f-*96Sy -aeWa 21. 2i'+20 + 2g. 21 — 20 - 2g. 2% + 20 — 2g. 21 — 20 + 2g. 2i + 2e. 2» - 2C. 2l'+2g. 28 — 2g. 2%. j ■ 7 — "002 5 — •014 + ■027 — •on + -003 i 3 + •027 — •021 + '°33 + -007 + •269 + '002 + ■066 + -iSl) ' J 1 + •002 + •062 — •069 -'°S3 + '335 + •501 + ■151 -•166 + -948 1 -I + -III — •030 + •162 -•150 -1-251 — •021 + •010 — -061 -1-693 -3 + ■050 -•048 — •005 — '054 — •OO3 —•362 — •016 — *026 1 ~s — •024 — •002 — "007 —•003 — "001 1 ~~ 7 Sum... — •001 +•138 + - 043 + '022 -•175 - "97o + 773 — -202 -•18S - -580 7 5 3 j ' — 1 • -s 7 162, Characteristic efk* a. Values of A. 2t+c + m + 2g. 2t' + c+m— 2g. 2i+c—m+2g. 2» + e— m— ag. + •0002 + •0065 + •0766 — ■0163 -■3091 — -1008 — "0028 + '0070 + -0457 —2-3608 — '2389 — -0729 — - 0020 + *000I +•0055 +•1939 —7781 + •0376 +'0530 + "0009 — -0003 ~ '°333 — -0382 + -1489 -1-7625 + •0011 + "0014 + -oooi 2» + c + m. — '0010 — -137-1 ■-5-9398 — "2096 + 1-2745 — -0282 — -0009 — -oooi — '0019 + -0678 + 1-3612 —4-0717 — 1-4012 - -0388 — -000 2 ■3457 -2-62IIJ •4871 -I-682S -5'=424 -4-cS.)ij 86 Mr. Ernest W. Brown, Theory of the B 21. 2t + C+m + 2g. 2J + C+H1 — 2g. 2i + c — m + 2g. 2i + C — m — 2g. 2J + c + m. 2J + c-m. 7 5 — '0001 -•0031 + 'OOO9 — '0010 + -O0IO \ — ■0083 -•1471 + -0045 + "1262 - - °943 + •0747 i -•5658 — -1190 + -0279 + '0681 — 2-9362 + •9469 —i + +864 -•3057 -1-5007 + I-8829 - T3 2 7 -•3019 -3 + '0646 + •0295 - '1013 - -1163 + 1-1756 -■8338 -s -•0059 + •0011 - -0178 — '0017 + -0249 — •0238 ; -7 Sum . . . + '0005 — -0003 + -0004 — •0001 1 1 -•0285 -•5443 -1-5877 + 1-9600 -1-9633 — •1370 w,,v.,£~ l -7-aee'k 2 a 2/. 21 + c + m + 2g. 2 i — C — III — 2g. 2i + c-i-m~2g. 2('-c — m + 2g. 2( + C-m + 2g. 21 — c + m-2g. — -ooi + -ooi 5 -•088 + •002 + -037 —•320 + •074 + '00 2 + -212 ! 1 + •004 ■-•142 — •21 + "014 + •005 + 1-054 _ T — -012 + •392 + ■268 + i'33 -•303 — -029 ~ 3 + •622 + ■004 -•047 + -056 -•193 — 'OOI — 5 -•051 — 'OOI + -ooi + •016 -7 '. — -ooi t Sum... +•562 -■155 + ■086 + 1-403 -•475 + 1-274 ] 21. 2J + c — m — 2g. 2J — e + m + 2g. 2i + c + m. 2i — e — m. 2J + c — m. 2i-c + m. 7 5 — '005 — 'OOI + '013 + 'OOI — '009 — -211 — -005 — -048 + -6n + "034 - -857 1 — I'04O - -687 -3-090 + I-549 + -850 -5 - 45 j -1 -i'3i5 + I-I37 - '698 - -482 +7-75 - -326 j -3 + -084 — -249 - -536 + -027 + -005 — '014 | -5 -7 + -ooi — 'OOI — -005 + -007 Sum... -2-486 + -195 -4-378 + 1-718 +8-647 -6-656 Motion of the Moon. 163. Characteristic k 4 a. Values of «7 21. 2t + 4g. 2i'+2g. ,2?.. , '. 7 " 5 + '0002 — "0003 i 3 + '0064 -•0497 ; 1 . . • + •0029 + •0228 + •2695 1 —-I + ■0658 -•2685 ; -3 . + ■0425 — ■0693 ! -s + •0025 — - 000I -7 Sum ... — - 0002 '•■•"••-..".. + 'II35 -•3085 + •2195 1 B 7 24 + 4S- 2g. 1 2i. I S + '0001 — '0003 1 3 + '0025 - -0051 J 1 + •0013 + - 2o66 —4-7553 •' ~~' 1 — '0176 — 2-6677 1 -3 + •0765 + -2237 i -s + •0043 + -0006 1 ~ 7 — 'OOOI i Sum . . . + "0645 -2-2343 -47607 %.„£"" '-f-ak 4 a \ :: ' ; 2«. 28+4g. 2*'-4g. 28 f 2g. 2i'-2g. 2i. 7 5 -•005 + '002 3 + -049 + -139 — '022 1 . + ■190 — •006 + 728 — 2'37I — 1 + "002 + •001 — '921 + 715 + 5'93i 1 -3 —•005 -•066 + "004 - "015 -5 — •010 — •004 -7 Sum — •013 + •235 -•997 + 1-588 + 3'5 2 3 Mr. Ernest W. Brown, Theory of the 164. Characteristic k 5 . Values of 165. Characteristic^^ 1 . Values of i A B : i. 2i + 3g + 2e. 2i+3g — 2C. 2i+3g. 21 + g + 2C, 2»' + g — 20. 2* + g. 1 1 ; 4 ; — •00002 — -OOOI3 — •00001 3 — '00038 — "OOOOI — '00002 -■OO554 — '00098 [ t 2 + '00002 — •O1872 — •00156 — 'OOI02 — '10446 — •02841 1 1 + •00132 — '61256 — '09289 + 'OOo8l + •00561 —•23460 1 1 + -02SI5 + •00677 — 3'348o7 + 1-66897 + '0333 T ! +'01907 | 1 i _I -■45559 — •O0667 + '04340 — '00167 + •16980 — -00656 — 2 + •00526 + •00524 — '00258 — -O0123 + •00455 + •03469 _ 3 — '00002 + 'O0O07 + -00064 + "00041 + •000 1 2 + •00074 -4 — - oooo6 — "00002 — -OO004 + •00003 Sum... — •42392 — '62627 — 3' 4 oio9 + I-6662I + '10326 | — '21603 A \ B i. 2 «' + SS- — -00004 ai + 3g. + •00004 it + g. 3 + "O0OO2 2 + '00002 — -00003 + •00027 1 •OOOOO — '00056 -■01230 + "00036 + •00672 -•01946 — 1 + ■00345 — '01322 -"00932 — 2 + '00221 + •00029 + '00224 -3 + 'OOOOI — 'OOOOI + •00009 1 % Sum + •00601 — '00677 | -•03846 ! [ 1 I •/ — is k .-f-ak 5 i. 2* + Sg- 2i + 3g. 2 J + g. 3 2 + OOOOI 1 — 'O0002 — 00147 1 + •00001 + '00067 gk-= " -■00883 j — 1 + ■00033 -•03317 + •02556 — 2 + •00258 + "0O02I + ■00035 i -3 — '00005 ■1 Sum + •00287 — •O323I + •02445 imp Up ■ Motion of the Moon. So v'-ICk.e.-HlkV 1 1 fli I ;:■■■ ■', ■; ' : ;■:■ '::'■■:. ■■ : ': : : 2l+3g + 2C. 2i + 3g-20. ' *'+3g- 2i + g+2C. 2J + g- 2C 2*>g. j 4 3 — -00001 — ■00025 — "00002 I 2 — -00079 — ■00003 — '00002 -•01397 — '00117 1 1 + '00003 — •07184 — ■00368 + '00012 + •00619 — •O2814 + •00088 + •1018 -•35528 +•18055 —5728 — 1 -•04391 + •01382 + '09779 + ■00192 + •01989 + •02079 — 2 + '00605 + •00076 + •00223 : + ■OB2O4 + •00019 + •00478 -3 — "00006 + •00011 + •00007 + '00004 '. -4 — '00001 :;\::K;:\--::;::y : :1 Sum —■03702 + "0437 -•25886 + •18468 -•5607 -00372 1 + •00875 166. Characteristic k 3 ee' '. Values of P m 1 I H HI ■ 1 J ■ ■ i, ':"■:' 2t+3g + c + m. 3i ' + 3g — 0— m. ^i+2>g + o—m. 2t' + 3g-c + m. \ 4 3 + '0004 1 2 -•0038 + •0001 + ■0004 j 1 1 — •6001 ' -•1546 . . ' • + - oo5 : 8 + •0232 | 1 ° + •0190 + •2873 + -0360 -■4097 i . — i- -■f'399 + •1250 + ■1364 -•3864 —2 -■0138 — •0023 — -0023 + •0074 1 -3 — •0015 + -0003 + - ooo3 ■^•0002 -4 + •0003 Sum — ■6360 + ■2523 + •1763 — 7653 Mi ml :' II 111 m ■■■"■' t.v 2i + g + c + ta. 2i + g-c— m. 2i' + g + c— m. 2i + g— c + m. : : ■ : : : : ;: : ■; ■ : y: ■/ ; : : : . ;y ;:y:j .4 — •0002 ! 3 + - 0002 -•0050 '. — '0004 — -0002 2 + '0026 — -1081 — "0099 + •0096 1 + -0793 -■5 6 i7 — '1619 -•3477 ' • + i"335° + '45 °5 ' -1-6058 -•7741 — 1 - -1471 -•0756 - -4687 +•3263 — '•% '"'.' ■ + -0033 — '0012 + -0T73 ' +'0129 .;:' -3 — '0012 + •0001 + 'oon •' ; + - QQ02 - :: • -4 — -0007 Sum ... + I - 27I4 -•3012 —2*2283 -•7730 iii§ go Sum Mr. Ernest W. Brown, Theory of the t. 2i + 3g + c + r ■ 3 2 1 + '0009 : ~ l "•1385 i —2 + ■0153 «_- t — •OOIO •1233 </=il -ak'ee' 2t + 3g— c — m. 2j + 3g + c-m. — 'OOOT — "0098 + •0880 — toSo — -ooi 5 -•0314 + ■0002 + '0O2 2 + •0353 + •0017 + •0003 + ■0396 2J + 3g-c + m. + •0013 — "1026 + •3498 + '0064. + •2549 2 1 O — I — 2 -3 Sum 2t + g + c + m. 2i + g — c — m. 2t + g + c — m. 2» + g — o + m. — -O002 + •0001 — •0078 — ■0003 + '0005 + •0050 —•2215 — •0109 -•1057 I +-3491 -•3813 -•5112 + •6618 i +'1272 ■ — '0260 +•4053 +•1364 | +'0025 — '0001 + •0088 + •0010 — •0001 + •0001 +•4838 —•6369 — ■1082 + •6940 167. Characteristic W 2 . Values of A B 1 i. 2t + 3g+2m. 2i + 3g-2m. 2t + 3g. 2J + g + 2m, 2J + g — 2m. 1 2S + g. j i 3 — *0002 —•0005 + -0003 1 2 — "0004 + -0003 — •0001 — - 000I — '0242 + •0091 I 1 + •0002 + •0082 + -0004 — •0023 -•3466 + -J547 | + •0244 + •0374 + •0082 + •0646 + •0539 -•0195 — 1 -•2550 + •0022 + •0993 -•0734 ■0000 + •0083 — 2 + •0279 — •0014 — •0180 — -0109 + •0008 + •0066 — •001 1 — '0002 + - ooo8 — •0010 + '0004 Sum... i — '2040 + •0463 + '0906 —•0231 -•3166 + •1599 Motion of the Moon. m i. 2i + 3g + 2m. 2l' + 3g-2m. 2/ + 3g. 28 + g + 2m. 28 + g — 2m. 2» + g. 3 2 y y-: : -:"v: r ^-: : / ::y: : v ■ y — •0011 + "0004 I + -0003 -■0473 + •0186 — I + "O02O -•3075 + -0047 + •13 + •0012 + •250 + ■375 + •117 -•l68 + ■11 — •026 glcV = — •0090 '•'• """? —•0286 — "0006 + •024 -■0015 + "0003 + '0009 -3 — -0003 + "0002 Sum... -•3344 + •13 + •275 + •290 — •II —■006 168. Characteristic ke 4 . Values of 1 A B i. j 2«' + g + 4C. 2i + g— 40. 2i + g + 2e. 2i + g~2a, 2i + g. 4 1 + -00086 + •00089 + •00018 3 + "00001 + •00139 + ■00066 +'01631 + -00642 2 + '00084 — •OOOlS + •02200 +-00715 + •12224 1 + •02860 + •00114 + •42.113 ^'00042 — •00718 1 +-55401 — •O783O — •06698 +"00067 + ■00061 —1 -•14499 — •OI869 — •00019 —'03848 + -00043 — 2 1 + -00098 — '00088 + •00010 — ' OI 347 — •00719 -3 1 + -00003 — •00003 — '00083 — '00065 —•00376 -4 — "00006 — '00045 — "00002 — •00018 j Sum... + •43942 -■O9469 + '37544 —'02802 + •11157 j "/ — is ke «-^ake 4 t. 2! + g + 4C 2! + g— 4c. 2i + g+2C. 2i+g-2Q. 2i + g. 4 + "00004 + - O0002 3 + •00021 + '00001 +-00070 +'00013 2 + •0000.1 + -00024 + '00044 + "00096 + "00495 1 + -00054 + •00389 + '0l6lO +'00193 —"00085 0:. . + •02001 — -00863 — ■OO723 — '0121 .0 = + '00027 — 1 — -01421 — -00073 — •OO085 —■OO473 —-OOI17 — 2 + •00114 — '00002 — '00015 — '00055 —"00099 —3 —•00005 — •00013 — -00001 —-00016 -4 — 'OOOOI —•00002 Sum... + ■00743 — •00500 + •00817 —'0138 +'00191 Royal Astros. Soc, Vol. LVIL 9 2 Mr. Erxest ~\X. Brown, Theory of the 169. Characteristic keV. Values of i. &i + g + 3« + m. 1 4 1 3 1 2 — •0018 J 1 — '0464 1 — •6708 | — 1 -•2353 1 ~ 2 + •0005 1 -3 — '0014 1 —4 1 — •0015 1 Sum 1 -'9S 6 7 2i + g-3o- + ■0027 + •0418 + •0191 — ■0314 + ■0736 + ■0479 + ■0023 + •1560 22 + g+e- 2J + g — C — I 2/' + g + c-m. i 4 1 3 — '00.10 1 2 —•0285 i I -•4227 1 ° — '2906 — 1 + ■0182 — 2 — '01 1 1 l " 3 -•0187 | -4 — •0010 j Sum -7554 + ■0011 +•0338 + •4429 +■0554 —•0390 + ■0263 + ■0275 + ■0011 + - 0002 + -0052 + 'I33 1 + 1-3679 + -0476 - -0529 + "0025 + -0054 + - 000I + •5491 + 1-5091 n/- !»» -akeV i. \ 2* + g + 3c + m. 2 j + g_ 3 C_ m , 1 4 + •0001 1 ■ 3 ■ + •0032 ! 2 + •0096 1 — -0012 + •0274 1 -■0373 + ■0200 — 1 -•0525 + ■0029 — 2 + •0002 + •0001 -3 — •OOIO -4 — •0001 Sum -•0919 +•0633 21 + 2 + 3c-m. + •0001 + •0044 +•0485 - -0205 — '0019 + -0306 2j + g + 3c-m. 2s + g-3e + m. — "0002 + '0002 — •0057 + •0074 + •0059 + •1638 — - oo6i + •8067 -•0580 -•0775 -•0865 + •0025 — "0071 0000 — '0002 + -0003 + -9034 -'1579 2! + g — o + m. •0057 •0866 •0704 "°°55 •0633 ••0729 ■•0051 —•3097 2» + g-30 + m. — ■0004 + •0023 + •0054 — '0192 -■0057 — •0002 -■or7S Motion of the Moon. y.? «/ — is keV -f-ake 3 e' {continued) i. 2i + g + c + m. 2i + g— e— m. 28 + g + c — m. 2i + g— c + m. 1 4 1 3 + '0010 +'0001 — 'O0O2 2 — "ooo8 + ■0310 +'0038 —•0056 I -'0253 + •0220 + '0886 — ■0219 — '0760 + ■0329 +"0157 +■■005:2 — I -■0152 + '0086 + "0450 -•0258 1 — 2 -•0059 + "00l8 +'0012 —■0052 — 3 -•0015 + '0004 — '0002 -4 Sum ... -""47 + •0973 +T548 -•0537 1 I70. Characteristic keY 2 . Values of A ■;'.*• 28 + g + 2C + 2m. 2S + g — 20 — 2m. 2i + g + 2C— 2m. 2i + g — 2C + 2D1. ■■/ 4 + •0026 3 + •0576 + -0007 — •OO39 ■:■■ 2 + ■0007 + •5520 + "0296 -'I33I v/ v I + '0020 —•0432 + -4848 - '0045 ■/ : —2683 +•0063 + -7659 + •0171 ■■■:■: "I -T694 + •0093 — '0082 — '2014 -2 + "0240 —•0005 — -0057 —•0288 -3 — •0304 + -0036 — •0014 —4 — •0024 Sum ... -•4438 +•5841 +1-2707 - -3560 :!■:. ■'■.■:."■:: 1 i. A B 2t" + g+iC. 2! + g — 2C. 2i + g + 2m. 2i + g— 2m, 2»' + g. 1 ' 4 — '0008 + '0003 — '0002 3 — '0003 -•°i57 + '0172 -•OO57 | 2 — '0109 —•1408 — '0191 + '358S — H83 • r —•1984 —•0073 — •7306 + 2-1857 -•7709 j ... •. -•2647 — •0081 -■0518 — - 0337 + "0233 | 1: — '0102 + •1564 + •0428 — "0030 — 'on8 I \ —2 + '0010 + •0163 -■1725 + "0199 + '1020 j : -3 + ■0139 + '0006 — •0174 + '0005 J + '008l 1 : ~ 4 + -0009 — '0006 Sum...; -•4687 + '0006 — ■9492 + 2 - 5457 j 1 -7735 | 94 Mr. Eenest W. Beowk, Iheory of the V-is te v»-r-akeV 2 28 + g+2C + 2m. 2l-t-2 — 2C — 2m. 2i + g + 2o — 2m. I. 4 3 2 I o — I — 2 -3 —4 Sum. 4--OOOI — '0264 -•2254 —•0252 -•0054 — •0001 + •0001 + •0027 + -0844 + •0735 + •024 + •0011 + - ooo6 + •0198 + "°934 + •17 + •011 + '0005 -■2824 + •186 + •30 2! + g + 20. 2i + g-2C. 24 + g + 2m. 2i + g-2m. — '0002 — ■OO76 — •0287 — •032 — •OOI + '0021 — •OOO7 — •Ol88 + '020 + •13 + - OI93 + - 0007 — •067 + -iS •316 + '5o 171. Characteristic kee n . Values of 2S + g-2C + 2m. — '0002 -•OI59 -•13 — *040 — •O280 — •0013 21 + g. + '0004 — 'OOOI 0008 + -0IS7 — "0048 0782 142 + •2968 + •089 -•0925 = + •0104 067 + •10 +•036 •0271 + •0027 + •0140 •0008 + '0004 —•047 i. i 2t + g + C + 3m. 2«' + g-c-3m. 2i + g + c -3m. 2i + g — e + 3m. 4 ! + -0004 1 3 1 + -0242 + '0024 + '0002 2 + -4352 + -0703 + •0057 1 — 'OOI2 + I+45 1 + -8945 — •0051 - -i5 l6 - -1946 + '3422 + •3416 — 1 + i'3342 — -0141 — '0203 -•2245 — 2 — -1292 — -0006 — •OI4I — •0671 -3 - -0256 — "OOOS -•0039 —4 — '0014 Sum ... + ro252 + 1-6956 + I-2745 + •0469 i Mnl',,,,1 ,./' //(, M,l,,i,. 95 illllllBIli t. 2i + g + c + m, 2s + g— c — m. 2£ + g + c — m. 2i + g— c + m. 4 — -oooi ^^^^^li^lii 3 — '0102 — •0013 2 + -0020 - T742 -•0316 — '0191 I - "O07S — •6600 -■3931 + '4424 ppwv — /6609 - -7S27 + 7341 + -8012 liipslfiiltppls — I - -4825 + - oo6o + •3980 + -2141 ■Rv — 2 + -H55 — -0035 + •0141 + -0480 ^^^^l$|||l||::-:-:;;'!;:; -3 + "0152 — -0002 + '0007 + "0022 -4 + "0006 — 'OOOI Sum... -~roi76 -i - 5949 + •7208 + 1-4888 |I«|ilSK«H»: i pit mm ^^ftlllll v'-isw.- r-akee'- 1 Hi' «'. 2i+g+c + 3m. 2i+g-o-3m. 2S + g+C — 3m. 2i + g— e + 3m. ^Bill: • 3 + -0008 llpl^ 2 + ■0341 + '0021 + -0003 pit/ I — - 000I + 7392 + •o'J34 — -0023 ^^811111= — -0367 + •1646 +•1380 -•3056 ^^Klllil: — I — 1-3081 — ■0031 +•0159 — •1227 — 2 - -i°53 — '0044 — •0054 -3 — -0023 — -ooor ^^|fSg:g:| Sum... -i - 45 2 5 + -9356 + •2150 -'4358 ■pir |I|p8iKRtti:»?S;¥ llllt WBmffMMimm i. 2t + g + e + m. 2t' + g-c— m. 2s' + g + c— m. 2i + g— c + m. 3 — -0003 ^tajiiiii; 2 + '0001 — '0124 — '0009 : —■00 II ^^^^^^^^0^y?/yM: 1 — ■0006 -•2550 -•0253 + •1352 '- — T712 + •6440 + '2329 . --6877 ^fclllll — i + "43IS + -0046 -•3420 +•0838 ^^Slfi ; ; — 2 + •0628 — -0002 + -0039 + ' 00 35 -3 + •0012 + •0001 + '0001 lilt/ iNiiii... + - 3 2 3S + •3807 - -I 3i3 — '4662 mt^^^lfS-^§^ 1 Mr. Ernkst W. IIrown, Tlmn-ji of tl»: \~i. Clitirtt'ii'i-ixt'u'V'e.a. A alue* oi 2i. 24 + jb'-' '■ 2i'--3--r-C. 2I + .i.'-r.:. 2.+ ?-.'. ^^^^^^^^^Iiiilljii|jli!|lii9^?ii|!l5s ..... . _ Bi|Sll|il§Ssll|ilM 7 I + -oooS llSiltp + -0002 — TOOI + 'OO20 + '0364 S|#|3S-; ; 1^B ^^^^^^^^^^^iBIiililiiiliSlilil 3 + '0015 + -0149 + "i5 86 + 1-1788 SlililSiltll + -0491 4- >Si2 +4' 6 i T 7 - -I4S7 'tifelC villi — i + 1-5291 4- -0369 - ^077 4- -0151 + '-57^ — -0321 + -i66S 1 1 — j + '0723 -5 ! — -0063 4- -ooSo + '04^2 + 006S ;';yMS?"--1:^i -7 ' Sum... + '0021 4- 1 -6480 4- -0003 + 'ooi 7 4-1-0563 +4-6669 + -0002 SiiipiliilSSfli^H + I '2090 v — 1.-. 1: Cj +-ak 3 e« '■ 2i. < 2i t 3g + e. 2i- + 3i;--c. ai + a-i-c- 21 rg- 0. 5 3 ' iiiiiiili + •0019 — 1 + -I54S -3 + M2S -5 '. + •0065 -7 ; + '0004 + - 0001 i Sum. + -292 + ■0006 + I09S + i7' — •°33 + •001 1 +•250 4. •0065 + •523° — I •264 — •090 + •0059 + •0001 •8(S + -0016 + i5°7 + -966 + - °95 + -0217 + •0003 + I-235 2(-i-3gim 5 ; + -0003 3 + •0020 1 + ■0645 1 : + -S9 2 3 3 : -•7884 5 ■ + •0212 7 ■ + •0014 173. Characterifitick z e'a. Values of Sum... •1067 2i+ 3;r-m. — -0003 + -0032 + '0619 -1-7294 + -35°5 — -0150 — '0007 -1-3298 21 + g + m. — '0003 — -ooSS — -42S 6 — 13-0974 + '95 > 5 + - 5°34 + -0257 + -0006 — 12-0509 + -oooi + -007s + -3012 + i*5 6l 7 -11-1432 — -H49 - -0047 — -oooi - 9-3921 Motion of the Moon. 97 sj — iz kv ,-~sk. s e'a 21. 2i + 3g + m. 2« + 3g-m. 2i + g + m. 2»' + g-m. 7 S — '0003 -+- '0002 3 + '0001 + ■0001 — -0292 + •0209 i + '0038 + •0034 —37306 + - 59 2 3 — i + '1406 —•4881 — -7886 + 9'45 28 -3 + 738S — -3000 + -2256 + •0016 -5 + ■0192 — •0088 + '0021 — '0002 -7 + •0001 — •0001 Sum + •9023 -7935 -4-32IO + 10-0676 1 74. Characteristic ke 3 a. Values of A Ill * 2s* + g + 3e. 2J + g — 30. 22 + g + C. ! 21 + g— C, 9 — 'OOO I 7 — -0089 — -0003 — •0031 S — 'O00I — -06ll — -0122 — - I02O 3 — -0141 + ■0071 - '399° — -4204 i -•5143 — •0045: —-4881 + - 0II3 — i + ■6224 + •0981 +-0182 — '0063 -3 + •0150 + •0219 —-0105 ■f -0946 -5 — '0028 + •0006 +-0304 + - OI59 ! -7 + •0029 + '0039 + '0002 ■■ :■■■■ . -9 + '0004 + •0001 Sum + ■1094 + •053* --8S75 — •4098 ■/ — is ke!>a -r-ake 3 a 2t. 2i + g + 3c. 2t* + g— 3c. 2i + g + e. 2i + g— e. 7 — •0004 — 'OOO 1 1 S — •0087 — '0003 -•0043 1 3 — •0003 -•015 --0157 -'0533 1 i — -Ol()l -•047 — -0554 -'°73 — i + ■0641 + •0113 +-iii + '034 -3 + •031 + '0009 + -020 + •0123 -5 + -004 + -0044 + -0007 -' ! + "0005 + - 0002 Sum ... j + •080 —•059 +-064 —•084 9 s Mr. Khm:s-i W. Ukow.v. T/.-onj of t'» 175. Characteristic ke 2 e' a. Values of A 28. 28 + g + 2C + m. 2i + g — 2e— m. 2i + g + 2c-m. 22 + g — 2C+m. 2i+g + m. 2i + g— m. 3 9 ! — '0002 + - OOOI 7 - -0074 + -0076 + -0018 — -ooio 5 ! + -0047 — -2310 — '0004 + -1880 + '°7'^4 - -0432 1 1 3 + "1891 - -5328 — -039 2 - '4235 + 2-0144 -1-0514 '■§ 1 + 47223 + '2560 - -6lS4 -3-1861 +4-S725 + -I2II B — 1 — 'i3 28 - \3295 — 27916 + -2076 - -0405 +5-5200 4 -3 - -4499 — -1708 — -2700 + -1003 + '2134 - '2235 1 -5 + "053* — -0058 - -0183 + -0035 + -0769 - -0745 : j -7 + -0137 — -oooi — '0080 + -0021 — -0024 1 -9 + -0003 — -0002 a Sum... + 4-4005 — I'02l6 -37431 -3-1025 + 6-9190 +4-2451 n/ — IZ teV .-5 -ake'Va 2i + g + m. 2i + g — m. 2i. 2t' + g + 2e + m. 2S + g — 20-ni. 2i+g + 2c— m. 28 + g— 2c + m. 7 — - O002 + -O002 m 5 + •0001 — •0170 + -0124 + -0022 - -0013 3 + '0050 -•2344 — •001 1 - -1335 + -1255 — -0708 '■'I 1 + , z7 I 5 -•2173 -•0391 + 2-7063 + 1-3020 + -0216 1 — 1 — •0299 -•O979 — ■8122 + -0885 + -0199 -4-6849 •M -3 + -4ISS — •0I08 + •2425 + -0070 + -0978 — -1008 m -5 + •0347 — •0002 -•0068 + 'OOOI + -0058 — -0052 m -7 + •0011 — •0006 + -oooi — -oooi Sum... + •6980 -■5778 -•6173 + 2-6810 + 1-5533 -4-8415 1 ■"''""^ °* H : " : " : H m :M 1 1 'vmm WA I M J Motion of the Moon. 99 CHAPTER VIII. Terms of the Sixth Order. Section (i). Formula and Methods of Procedure for u. ij6. Terms in only two characteristics, e 4 k 2 , e 2 k 4 , have been calculated ; for those in e 6 , in which very small divisors do not occur, the elliptic values can be substituted ; those in k 6 are insensible. Xo terms with arguments 21 are calculated, as no small divisors are present ; the constant C enters only with these terms, and it may therefore be neglected. The method is that of Chapter VII.. with Q 1 = 1/ = A' = o. Also a takes the values 6, 5, 4, 3, and b = 6-a. For u 3 u 3 , s 3 s t , z 3 z 3 , Dz 3 J)z 3 , u 3 s 3 + s 3 u 3 = 2u 3 s 3 , &c, we must substitute the halves of these functions when dealing with the general formula). 177. The main difference arises in the development of Sf. S(l)f ). Here *i 8 £4 , dii , , ,B 2 f9 , , ,3 2 f, , of, , of, 00 og oV eg- cc eg with a similar expression for S(i)f ) ; o 2 f 2 /3c3g, o 2 (Z>f 2 )/8c3g arc zero. For the first two and last two terms the formula) of § 134 are available with a = 4, 3, 2 and b — 4— a, a=2, 1 and b=2—a ; when a = b the remark at the close of § 176 must be noted. For the other two terms I find, for the coefficients of £ ±(2,+1 ' c > only, 8 2 f, do' 6c' ?■■■£.. ox* d 2 P^ = ( 8M l'" 8 + 8 k» M o) » g-^(-D f/ 2) = -(8M k ,S -%,M ). No general formula) for the derivatives of Q were obtained, owing to the difliculty of expressing them in convenient forms. The cases are a > b = 3, 1 j characteristics ke 2 , k ; k 3 , k ; a, b = 2, 2 , „ ke, ke ; a, = 1,1, „ k, k. Royal Astrox. Soc, Vol. LVIL p 100 .Mr. KkM.M W. !>:;.. w.v. Ti>- •r<j >f the For these cases we have, using the notation of § 134 for 3" and of § 135 for :', /i Q = 4 2'^2)( ! :. l = t )- 2 a'^. 1 2>,. 1:> M §1 a = ke 2 or k 3 , : b = ke, I og J og dc 2 S'™ s \ e )-2 5 ' ke Z? 5k0 ; : 6 = k, : ,-^-r =2S'^(5 S k )-2S' k 2)s k , dg ;■■-■«: j Vdg/ 3 2 Q where (^ ta ) K+c denotes that the terms whose arguments are ±( 2 i" + g + c) only are to be used, and similarly for (y k e) B ^c 178. The calculations were made on the plan outlined in Section (iy.) of Chapter VIL, with certain abbreviations. As in most cases only three significant figures were necessary, it was found possible, in forming the products of series, for the computer to add each pair of logarithms from the slips and look out the number corresponding to their sum from a four-place table (printed on a card) without writing anything down (see § 140). Only this last number was actually written down, and thus the sheets on which the various series were added together could be entered straight from the multiplication slips. The great majority of the products consisted of numbers with one or two significant figures, and long practice has made us so familiar with the logarithms of numbers from 1 to 99 that a glance fit the tabic was rarely necessary for these. Nearly all the other operations were so arranged that the use of logarithms was not necessary and the computation sheets were much abbreviated. In the few cases where logarithms had to be used they were written down on a spare corner of the sheet, so as not to disturb the general plan. Only those sets of coefficients corresponding to the arguments 2z±(2c-2g) required the special method of § 138. Section (ii). The Homogeneous Equation for z. 179. As the calculation of B t (see § 139) would have been long, the homogeneous equation (8) of § 7 was used. The terms calculated were those with characteristics k 5 e. k 3 c 3 . ke 5 , so that we have i2 t =o. The equation is then Put u'^utr 1 , *■'=&£, and divide by & The equation may be written f" + 8f"=(i) a — 1 - 2m- S -m 2 )«M'— 2(D + 1 +m)%Du' — hnhs'^—o, ■ ! 1 ins (1.) Wk If Motion of the M< oon. IOI or fact, (2) 2 -g 2 )su'~ 2(I> + g )zDu' + (g ' z — i -2m— im 2 W+2(g -i— m)sZ)w'— ?m 2 ss'^ 2 =o. (2) 2 2 The constant coefficients in the last three terms of this equation are small. In g 2 — i — 2m— 5m 2 = — -00044, go— r— m = + "°043 2 . 3 m 2 = + -009805. ... (3) Also, since Jhi ' has the factor in 2 , the term of principal importance in the determina- tion of ~ 6 i*> the first. If then the operation (V.^ 2 — </„ a ) _1 he performed on the equation , the first approximation to z 6 is given by "~ -* + U-go B>- gl * )~ Du + i+2m+ _m 2 -g 2 D*-g i>--go 2 2 where wc neglect ~ G on the right. I: 1 or the second approximation we substitute — ~6 (1) ( M o' — a ) (§ J 3 2 ) f° r tne nrs t term on the right and c 6 0) , lht ' for c, //«' in the other terms ; and so on. 180. As in Chapter VII., c , g are to be used for c, g in performing operations involving D. Let Sf" denote the parts of (1) due to c — c„, g — g (1 . Then 8f"=c 2 + +g 2 -±- +c 4 -J- +g t -J- ; do dg do dg the terms involving second order derivatives are found to be negligible, owing to the smallness of c/, c 2 g 2 , g 2 2 . Let q denote an index of £ in v.'. Then, using the notations of §§ 134, 135, 8 *"=2'^£ j 2D(zu')-2zDu'\ - 2(D+ 1 + m)s^'^u'. oc do l J oc Since 1+111— g is small enough to be neglected when multiplied by c — c„. g—go (there are no small divisors), this may be written : do do do with a similar expression for f}f"/dg. Hence, after putting c = c l): g = go hi the coefficients of £ in (4), we must add 1 sf" ■0 s -go 8 to its right-hand member ; this expression separates into two parts, involving the operators (J) 2 — go' 2 ) -1 an d (-^~~&>)~ l respectively. 181. The actual computations are comparatively short. The products z a u\, x a Du\, with a, b = 5, I ; 4, 2 ; 3, 3 ; 3, 1 ; 2, 2 ; I, I, are obtained according to the plan explained in § 140, and the other remarks there made apply here also. Moreover, as there are no monthly terms, small divisors are not 102 Mr. Ernest "W. Beown, Theory of the present. In most cases the terms multiplied by (3) can be altogether neglected ; when they are not quite insensible a simple inspection shows what additions arise from them. It was also found that a second approximation to the value of z 6 was not necessary. Section (iii). Values of A x /ar\, B^/aPX, ?/ A £ _1 / a \ t^/aX. 182. The tables giving the characteristics and arguments calculated are (see § 148) : — §• \. Arguments. I 183 184 e% 2 e 2 k 4 2l±4C±2g, 2t±20±4g, 27±4C, 2i±2C±2g, 22±2C, 2l±2C±2g, 2?!±4g, 2l±2C, 2l±2g ! 2Z±2g | §• X. Arguments. 1 185 186 187 lc'e k 3 e 3 ke r > 2i±5g±c, 2i±3g ± 3 c > 2i ± g±S c ! 2i±3g±c, 2i±gdto 2*±3g ±C . 2i±g±^c, 2i±g±3 c > 2i±g±c 2t±g±C The terms for which the method of § 138 was necessary, owing to small divisors, were Arguments. ±(20 -2g), Characteristics. e% 2 , e% 4 . I" §§ l8 3> l8 4 the arrangement of §§ 149-163 is followed exactly. In §§ 185-187 the final results for «^/a\ are alone given, as there was no definite stopping place in the computations, and the first approximations are the final results. i m i ■ Motion of the Moon. 183. Characteristic e 4 k s . Values of A:- J 03 » 2l' + 4C + 2g. 21 + 4c - 2g. zi + 40. 28 + 2e + 2g, 21 + 2C — 2g. 21 + 2C. 21 + 2fT. 3 — "OOOg 2 — •002 — •001 + "002 — ■O233 — 'OI4 1 + •002 — •045 + ■007 — - 02I -■OI57 + •101 -•371 + •005 -■O36 + '354 -707 -•I346 + •052 + "°59 — 1 -•079 + •008 — •052 + -I 33 + ■0089 — "019 -■015 — 2 + "OIO — 'OOI — '004 + ■001 -•0674 — •003 — "004 -3 — - 006 — "002 — ■OO25 — ■008 -•O05 Sum... — '062 -•082 + ■304 -'594 -'2355 + •123 -'35° B lijstllStt//. : 8. 1 21 f 40 + 2g. 2i+4e— 2g 21 + 4c 2i + 2e + 2'g. 28 + 20 — 2g. 2! + 20 2t + 2g. 1 3 + "0002 Ililtliii®-* 2' + •0072 + •001 — ■004 1 ; I + '012 — "OOI — "020 -■°493 -•066 + •084 1; O — •031 -•°73 —•179 + •142 — ■O054 + ■104 -•134 I — I + '027 '000 +•043 -■058 — '0037 — '010 + •003 BfcBlvi',' — 2 — •007 + •003 + - 002 + •003 + '0196 + •004 — •O08 WSll0iMM> . -3 Sum... + '001 — •002 — •OO05 + •004 + "002 ||pt8;SM¥:»::"'' ■■ Illiiiiwiv'. •' —'on -•058 -•134 + •065 -■0319 +•037 -•057 u e t V £ J -f-ae *k 2 'if 2i + 4Q+2g. 2i— 4e— 2g. 2j+4c— zg. .'2t-*4e+zg. 22 + 4c. '.21 — 4c. 3 — ■001 — "OOI ■2 + •002 + ■003 — 'OI :-I . — r oo6 — '00 3 ■ . •. — '01 + •001 — •001 : O + •010 + '011 -•044 +•031 + ■059 + ■027 1 -. -•028 + "001 + '02 —'007 "026 — "OOI — 2 + •010 —•004 + 'OI -3 •00 — "002 1111 ... — "or +•008 —•03 + •02 + •04 + •01 ■ 104 Mr. KliM.ST \Y. IJkow.v. rin,.,;i -\ lb, w . k ,£ ^-f-ae'k 2 (continued) r :•: : :-KvMvM ; : y>--:-y/x >■ m l. J 2! + 2C + 2g. 2» — 20 — ag. 2J+20 — 2g. 2S-2C-f2g. 21 + 20. 2t-2C. 2* + 2g. 28 — 2g. "ik 3 1 —•ooi -•003 + 'OOI 2 + '001 — 'OI — - O02 -■015 —•001 — •OO4 — "O06 I 1 + -oo6 i + '024 -•025 + •032 + '022 + -OI - -078 -•06 ■ ■■.■■.\:«.;j O — •145 — '022 + ''4 -•58 "-•056 + "I02 + •086 + •064 ;| — I + •083 + •007 + •003 ,+ ' 01 7 •OO + 'OOI + :o3 — ■Ol6 — 2 'OO H-'OOI — •OO4 —•004 — '002 + •004 + •003 :J§ -3 4- 'ooi — •OO4 — •002 J Sum... -•06 •00 +■61 -"55 — •04 + •11 + ■04 — •01 J 1 84. Characteristic e 2 k\ A r alues of . A ,. . . i. j 2i + 2e + 4g. ^:7:;:7;:;7::;:;x;;:::;:;::;;:;:;;V:7; 2J+2C — 4g. 28 + 2C + 2g. 2J + 2C-2g. 2% + 4g. 21 + 20. 2l'+2g. 3 ! + '0001 2 + •003 - "0051 + 'O0I — -ooi - '003 1. IS- ••//• \S6D.; + •178 + "002 - -3548 — '004 + 'OIO - -058 fe^: : :- : Mr^ + -8IO + •039 — S -I 477 -•063 + I '696 — 5-159 — 1 — •002 + •003 -•I38 - - 5573 + ■299 + -506 — 1-219 — 2' , +'017 — - 002 + - OIO - -0056 + •027 — -oio :■ 4r '-:00 : '% -3 — •002 — '0001 — •001 — 'OOI Sum.. + •015 + •992 -•089 —6-0705 +■259 + 2'200 -6-478 2 I — 2 -3 II 2J + 20 + 4g. 2S + 20— 4g. 2J + 2C + 2g. 2J + 2C — 2g, — -001 — -oio Sum...! +-045 2 j + 4g. + •436 — -406 ■■1399 + •283 + •006 — •0028 — - 00I + •002 — ■008 -•1373 + •004 — "014 + ■369 -•665 +•0896 + •267 — i'66i +•058 4- -268 -■0743 + ■013 — -148 + •001 — •001 —•0148 — •0003 — •001 + '027 — - 002 -'■;w 2J + 2g. j 003 -<5 062 004 oo:: -■■/).; ' + Mil saHT Motion of the Moon. 105 • 2J + 2C + 4g. 2i'-2C-4g. 2/ + 2C — 4g. 2t-2C + 4g, 2t + 2C + 2g. 2! — 20 — 2g. 3 *00 — ■OOI . 2 + •005 + ■001 •OO 1 — ■042 — •02 + •002 + - O0I — •O29 + 'OOI —•030 + '334 + ■003 + •272 — 1 -•073 -■08 — •O36 + *002 — 2 + •010 — ■001 — - OOI — 'OI -3 •00 + '0O2 Sum... + •01 -•04 — •12 +•25 -•04 + - 24 ('. 2i + 2e — 2g. 2i — 2C + 2g. 2» + 4g. 2t-4g. 2'i + 2C. 2i — 2C. 2 S + 2g. 21 — 2g. 3 — "OOI 2 — •002 : — -002 •OO — 'OOI — '029 1 — -182 — '2IO —•001 + -065 ;." + '661 + -o8 — -OI5 ~-i6 - -98 + 2-SS — - 002 -'353 + •443 + 1-725 -1-818 — •226 — 1 - -117 - -o S 8 + - I44 -■005 -•19 — '009 + 78 — '002 — 2 - '013 - -003 — "02 — ■006 — '021 — •ooi -3 Sum... + "OOI + - OOI — 1-29 + 2-28 + •12 -■29 + •25 + 179 — I"07 -•42 io6 Mr. Eknest W. Brown, Theory of the 185. Characteristic k 5 e. Value of y/—iz Ve -r-ak b e. i. 2i + 5g + e. 2i+Sg-c. 2t + 3g+0. 22+35-0. 28+g + C. •At+g—c. 2 I — *0I — 'ii O — "01 -2 '45 + ■82 -9-81 — I + ■08 -■07 + -6i + T4 + "24 — 2 + •01 + •01 — ■01 ' .""-' " 0I .' Sum... + •01 +•08 -■08 -1 -85 + ■96 -9-69 m m m 186. Characteristic k 3 e 3 . Value of ^/— i2- kV -r-ak 3 e 3 , i. 2i + 3g + 3=- 2i + 3S-3e- 2 j + 3g + c. 2i'+3g-C. 2/ + g+3C. 28 + g-3C. 2J + g + C. 2»" + g -0. 1 3 — ■01 2 — "01 -■03 1 + •01 — •10 + •09 — •02 + ■06 —25 + •09 + •12 -•l6 -•03 -•19 —1 -•°3 + •01 + •10 + •03 -■°3 + •02 + "OI -•04 —2 + *02 + •01 -3 Sum... -■03 + •07 -•13 + '02 + ■09 -•09 — •01 -•25 187. Characteristic ke 5 . Value of xZ—iz^-i-ake*. i. 2i + g+Sc. 2i + g-Sc, 2« + g+3c 2i + g-3c 2i + g + c. 2» + g — C. : 3 + ■001 + 'OOI 2 + •001 — ■001 + ■004 •000 1 + '002 + •010 + ■003 — •002 •Aod + ■010 — •004 — ■007 — •002 — •001 —■005 '• . — I*' — -009 + •002 — ■001 + •001 1 a + -004 — •004 + ■002 —•001 — •001: ; -3 Sum... + •003 -■003 + "009 — •003 + •002 — - 006 111 m m m 111 sill n ■ i ill I sill ■ Sill 1 iH Motion of the Moon. 107 CHAPTER IX. Results in Polae Coordinates. Section (i). Foil aula for Transformation. 188. Longitude.— The formula are given in § 41. The value of V is obtained by special values from equation (48) and the other Y^ from equation (47) in that section. By development, using the notation of § 139. we obtain 2lVj = («i-8,)-4-(m , s ), 2l V 2 = (ws-Sj-iV + la, 2 )-^, s ), 2.V, = (M-M)-^(« , So ), 2lV 5 = (M 5 -S s -?' 2 M 3 + g 2 S 3 -?i 1 M' + S I M')-r-(M , S ), 2 * V 6={M6-««-(«|«« + tt,«4+K*) + («l«« + V4 + W)+«l , M'-«, , M' + «»N-g,N}-l.(» 0> B ), where M J .(»„) = ^_!« , i''i+*j , i «iS L V ■ { 0,; «„ 2 ~v~ + i^-^ M'-=-(« 0) ) = M-f-Ko-r^+VrA!. Nh-KO = -^-+ < M- M !S All the products and most of the sums in Y„ V 2 , Y 3 , V«, M, M', N, were at hand. For V, the product u 2 u 3 was available ; the only products to be formed were, therefore, «j ?h by j/V and M'~(m 0; ) by V^o- In Y,. the known factor ?/, m b + «, m 4 + 1h 3 s was multiplied by i/< Ml 2 /V by M'-=-(« .) and u 2 /u, by N-=- («„,). In all cases the corresponding functions of ,<■■ were obtained by putting i/£ for £ in w, 189. Parallax.— Yot convenience in obtaining the latitude, equation (49), § 42, for the parallax was computed in the form ">.= ! + (/•») +(p°)+(e?) +(p?) r \ r ' 1 v r ' 2 ^ r ' 3 > r ' .] for orders up to the fourth inclusive. Here (£?), = -l(«i+»i)-K«oi»o). (7), sss {-K«»+«*)+K«i*+*i s )+i» 1 «i-i2 1 ! }-f-(Mo,«o, Po), -H w o> So. Po). s 1-3 1*2 - 1 " *-IS(«i*«, +«,»«,) -T 3 «(2ttl«^i+2« 1 » 2 M,+« 1 S» J +«,^) + | Sl *(« 1 + g s ) + 3 Sl - J („ 1+8l ) +1 ^„ i 4 + All the products and, by suitable rearrangements which differed with different characteristics, many of the sums were at hand. Royal A.stkon. Sue. V<>;.,. LY.1I, *4 U>- ?tll\ Ml£\i:sT W. Hli'iWX. 7'//"./ 1 // <;/' //'(" For the terms of the fifth order, in which the characteristics e\ e 3 k 2 , ek 4 were alone needed, it was shorter to use the Jacobian integral, equation (5), § 7. Here ^=0 and C is not present in odd order terms, so that All the products had been obtained in finding ?/,. An examination of the errors produced by using c , g for c, g showed that they were insensible. The terms factored by m 2 are also insensible in most cases. The order of accuracy for the parallax is found by dividing the characteristics required in the latitude by k. 190. Latitude.— To the sixth order inclusive this is given by (§ 43) p ft ft" r p° p J y ( V5i "R \ i=l x i=l Po ' where R, = (£») , R 2 = (J») o +1'^, R 3 = (£°) 3 + e«iS 2 -i~i 2 K+*i)}-^( w o. «o. Po). PO W-5 \ r ' 5 \° '5 = (') +A.W^-(^) 9 }.+- • s - L !{i a «^-A?!(5 I +?)}- \r/5 Po Po L Po Po vw o s o / J The expression for (,cr/r 3 ) 5 in R„/p is obtained by multiplying equation (4), § 6, by z, putting il x = o and neglecting the terms factored by nr ; we can also put c = c , g = go in this expression. All the products in the B,_; were known except those hi the last term of T? 5 /p . When the R,-_, have been obtained the multiplications by z,/p a were straightforward. Section (ii). Change of the Arbitrary Constants. 191. The usual constants c, 7, a used in the lunar theory are so defined that the coefficient of the principal elliptic term in longitude is that of the principal term in latitude, 27— 2ye 2 — Jy s + -3%7 c4 > and a is defined by the equation a 3 n i = JE+M. The results of the transformation to polar coordinates furnish the following coefficients for these two terms : — + •99972 871 e + -oi4S7 76 e 3 --o243see' 2 + 1-00499 9 ek 2 --oi7i ea 2 + •0021 e 5 — -046 eV 2 +'37i4 e 3 k 2 + -23 ee'% 2 +i'SS 3 <&*' Motion of the Moon. 109 and + 1-99974 473 ^ + -99167 5k 3 — -25024 2 ke 2 +-ooo73 k«' 2 — "0098 ka* + 1-48 k 6 — -795 k 3 e 2 + -o k%' 2 --0287 ke 4 --os keV 2 ; also (§ 44) a = "99909 31420 a. 192. Equating these, I find e= + 2-00054 273 e — -36681 52 e 3 + -o4873 ee 12 — 2-01160 2 ey 2 + -o342 ej — ) + ■049 e 5 + "3S eh' 2 — -246 e 3 y 2 — -56 ee' 2 y 2 + -9ii ey 4 , k= 4-i - oooi2 765 y— -49609 1 y 3 — -49924 3»/e 2 — -00037 ye /2 + -oo49 7 ("7) —■128 y 5 +ro7 y 3 e 2 + -o yV 2 — -095 ye 4 + -]2 yeV 2 , a= +-99909 314 a, -= +1-00090 768-. a a Section (iii). Numerical Values of the Constants. 193. The following are the values of the constants used in reducing the results to seconds of arc :- — m=i73 25594"-o6, »'=i2 95977"'4i5 , m =-08084 89338, e = -05490 056 , e' = -01677 191 , y = -04488 716 , 34i9 //, 59 6 > ^ = 8"-78oo. E M 81-500 . a E—M ll = a'-J+l/ = ' 002SOS32 The value of c corresponds to «i coefficient 2 2639' /- 58o of the principal elliptic term in longitude : that of y to a coefficient 1 8461 "-480 of the principal term in latitude ; that of a to the value 342 1"- 700 for the constant term in the sine of the equatorial horizontal parallax of the Moon. Section (iv). Numerical Values of the Parts of the Arguments and Coefficients arising from the Various Characteristics. 194. The coefficient of each periodic term in longitude and latitude is of the form XP (<r, c' u , y~, af). where X is the characteristic of the principal part and the factor of X is a quadruple power series proceeding according to powers of e 2 , d 2 , y 2 , af, with numerical coefficients ; in the sine of the parallax each coefficient is of the same form with the additional factor i/a. In the longitude and parallax only even powers, and in the latitude only odd powers, of y are present. In this section will be given the coefficients of all those periodic terms for which P is not limited by the calculations to a single term. The part due to each characteristic in. a coefficient of a given periodic term is separately shown. Those periodic terms which have had the parts due to the principal characteristics alone calculated are not set down in this section, since these parts are the final values for these coefficients given in the next section. The characteristics in all cases are composed of the new constants e, c, y, a u that m : lO Ml-. tliM.ST \Y. ili.-iWS. Tlf.r-' <; !/■■ is to say, they are the same as those of Delauxay, with the exception ai for a. A direct comparison of each part with the corresponding part given by Delauxay is thus possible, when allowance has been made for the slight difference in the numerical values for the constants used by Delaunay and myself, and for the change from a to a 2 . 195. The following table gives the various parts of the annual mean motions of the perigee and node * due to the separate characteristics set down in the first column. Annual Mean Motions. Char. Perigee, ': Node. 1 + 148524-92 1 i —69287-90 e 2 - 5 r[ r3' — 616-09 e n + 156-27 - 25-46 ■■■■.»"■; - 1739-85 + 260-59 «i* + 2-24 I IT e 4 + -o 4 + '• "07 e 2 c' 2 - '99 - '57 eV + 672 I ■*- 1-70 e' 2 y 2 — i'6i ! + -08 r 4 1 ■- I'S 1 + -05 Sum... ! + 146426-92 ! —69672-04 196. In §§ 197-262 are given the coefficients of those periodic terms in longitude, latitude, and parallax which contain more than one characteristic, The arrangement is primarily according to the orders of the principal characteristics, each principal characteristic being attached to a definite set of arguments which differ only by multiples of 2D and in the signs of the multiples of I, /', F. The notation for the arguments is that of Delaunay (§ 10). In the first column of each Table is placed the characteristic, and in the suc- ceeding columns the coefficients corresponding to the multiple (i) of 2D placed in the first row of each Table. In the last row is given the sum for each column, and therefore the final value for the coefficient of each argument. The coefficients in longitude are given in §§ 197-225, in units of o"-ooi. The coefficients in latitude are given in §§ 226-248, in units of o"-ooi. The coefficients in parallax are given in §§ 249-262, in units of o"-oooi. All coefficients have been calculated so as to be correct to the last figure given with the exception of those depending on e 6 , for which the elliptic values have & been substituted. This remark applies also to the Tables in section (v). * I have obtained the complete theoretical values of these two quantities and compared them with their observational values in the first of the two papers referred to in § 147, m m in 111 iH mm ■A-mm $■ m #111 Hi 11 m win m fill Sis |jj ill mm mm 111 m H im llill ♦1II1 Sill fill 'mm mm mm ■ till 111 'mmm m ■ ■ Motion of the Moon, in 197. Arg. 22 - D. .... i. 1 ' 1 3 1 . -. 2 i I ; + 49 + 8740 +2106246 e 2 .,—$- I -|- 67 + 5217 + 298973 e' 2 - 3° - 1993 7 2 - 3 - 309 - 31435 a, 2 + 2 ' + 37 e* + 17 + 433 - 343 eV 2 - 16 — 191 e' 4 e 2 y 2 - 3 — 13 6 - 13" e'V + 35 r 4 + 1 — 121 e 2 a, 2 + 4 7 2 ai 2 — . . . '2 Sum ... + 1 +127 +13902 + 2369899 • 198. Arg. I+2/,]). i. 3 2 i — 1 —2 -3 -4 e + 12 + 1446 + 174865 + 22648107 — 4608089 -35 221 — 291 ■ — '2 • e 3 + 10 + 574 + 20813 — 8 533 + 1231 - 4i43 — 114 — 2 ■ ec' 2 - 6 - 238 + 1586 + 77 + 2 1 ey s — 1. - 58 - 3314 + 18897 + 811 + 13 ! ea y 2 + 4 • 9 - 5° e b + 2 + 42 - 44 + 5 + 4 + 4 - 8 1 e 3 g /2 — 2 — 21 + 13 + 8 + 1 , e'V 2 - 17 - «5 - 19 + 38 + 4 ce' 2 y 2 + 6 + 15 - 2 e/ — 2 - 26 Sum... + 23 + 1979 + 191954 + 22639579 -4586438 -38428 -393 -4 ii2 Mr. Ernest W. Brown, Theory of the 199. Arg. V + 21J). i. 3 2 1 -1 — 2 -3 e' — 1 — 180 -21595 —659271 — 152090 -1255 — 10 eV — 2 -"3 - 3476 - 15490 - *5"5 - 636 -13 e' 3 + 1 - 5 + 11S + 3 e'y 2 + 10 + 651 + 57°2 + 1651 + 38 + 1 eV 7 eV — 1 — 10 - 26 + 27 + 3 — 44 - 3 eV 3 + 1 — 22 + 8 + 1 e^e'y'* + 4 — 2 + 120 + 82 + 14 + 1 e' J y' 2 — 2 e'y* + 2 - 5 + 4 Sum... -4 — 289 -24451 —668944 -165351 -1879 -24 200. Arg. 21J). 2>\. 5 3 1 a l + 8 + 735 -125394 e i a l -3 -383 - 2433 e' 2 a, + 7 — 21 y z «i + 42 + 3°4° a[ 3 — 1 e 4 a. — 1 - 9 + 5 e 2 y 2 a 1 + 10 + 36 7 4 «l - 17 Sum ... +4 +402 — 124785 201. Arg. 2 '+2Z'D. i. 3 2 1 -I -2 -3 -4 6* + 2 + 169 + 13241 + 771167 — 212622 -31054 -531 -7 e* + 2 + 56 + 1478 - 1038 + 92 - 31 - 59 — 2 eV 2 — 1 - 25 - 59 - 69 + 36 + 3 eV - 8 - 299 - i°37 + 95° + 279 + 16 e 2 ai 2 — 1 — 2 — 2 e e + 1 eV - 3 - 9 + 3 — 2 + 1 e*y+ + 1 15 5 — 1 Sum... + 4 4-213 + 14387 + 769021 — 211658 -3°773 -57° -9 IfJUl mm mm Ijjj ■ Marion of tin: M,n,n. * 13 202. Arg. l+l' + 2iT>. i. 2 1 — 1 ■~2 -3 -4 ee' -37 — 2662 — 110214 — 206896 —4088 -55 — 1 eV -16 - 356 - 660 — 149 - 393 -19 eg' 3 - 84 + 87 + 6 ee'y 2 + 2 + 94 + "54 + 740 + 79 + 2 ee'a x % — .1 — 1 Sum... -51 -2925 — 109804 — 206219 -4396 -72 — I ■ 203. Arg. l~l' + 2iT>. i. 3 2 1 — 1 —.2 -3 ee' + 3 + 216 + T3634 + T49260 + 27878 + 578 + 8 e 3 e' + 2 + 74 + ii77 + 61 + 3° 2 + 74 + 3 ee' 3 - 13 + 104 - 54 + 1 ee'y 2 - 7 - 203 - *549 + 385 - 16 ee'a, 2 + 2 Sum... + 5 + 283 + H595 + 147878 + 28511 + 637 + n 1 1 204. Arg. 2l' + 2il). i. 2 1 0':': — 1 — 2 -3 e n + 1 — 12 -7313 —7602 — 107 — 1 eV 2 ~4. -182 — 260 - 586 - 48 — 1 e'* + 2 + 6 + 5 + 66 + 66 + 4 Sum... -3 — 189 -7505 -8116 -151 — 2 114 Mr. Ernest W. Brown, Theory of the 205. Arg. 2F + 2^D. i. 3 2 1 -1 — 2 -3 r 2 -39 -4193 — 409912 — 56040 -53 eV — i —42 — 1614 - 849 + 754 + 68 + 1 c'V + 4 4- 1 + 57 y* + I + 54 - 834 + 48 + 9 yW — 3 e 4 y 2 - 6 + 4 4- 11 + 4 eV + i + 4 - 3i + 6 + 1 Sum ... — i -85 -5741 — 411614 55174 + 25 + 1 206. Arg. 1+ 2i x D. 2!,, S 3 1 — 1 -3 -5 ea^ + 1 4-29 -8546 + 18757 4-3180 + 5 e 3 a[ — 1 -38 — 127 + 214 4- 108 + 11 ee'^a. + 6 + 8 - 13 — 1 «7 2 «i + 7 4- 226 — 425 - 69 — 1 Sum ... — 2 — 8441 + I8554 4-3206 + 14 207. Arg. I' + 2iJ). 2i,. 5 3 1 — 1 -3 -5 e'a. + 1 4-112 4-17654 + 593 -90 — 1 e 2 e'a l + 1 + 43 + 599 + 1 . + 27 e' 3 a! - 3 + 3 e'y 2 a 1 - 5 - 258 - 38 - 3 Sum ... + 2 4-150 4-17992 + 559 -66 — 1 208. Arg. 3I+21D. i. 2 1 — 1 — 2 -3 -4 e? + 17 + 983 + 36339 -13273 — "97 — 296 -8 a' + 5 4- 106 - 95 + 13 + i — 2 — t cV 2 3 — 8 ~ 5 — 2 + 1 e 3 y 3 — 1 - 26 — 112 + 7 2 + n + 4 | Sum . . . 4-21 4- 1060 4-36124 -13193 — 1187 -293 -9 Miitiml !•/ I hi: .!/-<. i//. 209. Arir. 2l+l' + 2i\). 1 1 i. 2 1 — 1 — 2 . ■ ~~ 3 -4 -5 -268 -7700 -8664 — 2762 -85 — 2 eV — 2 - 33 - 45 - 6 - 8 — 3 eV 8 - 6 + 2 eVy* + 11 + 92 + 26 + 23 + 2 Sum . . . -7 — 290 -7659 -S63S -2743 -91 -3 210. Arg. 2I—V + 21D. i. 2 1 — 1 -2 -3 eV + 26 +'III2 + 9833 -^-2601 + 35 2 + 12 eV + 8 + 9 2 - 13 + 3° + 3 + 2 e 2g/3 — 1 + 6 — 11 + 1 eVy 2 — 1 — 21 — in + 85 + 4 Sum . . . + 33 + 1182 + 9715 + 2497 + 360 + 14 211. Arg. I+2V + 21D. i. 1 — 1 — 2 ~3 ee' 2 — 1 -13 — 1 1 70 - 9 + 10 -7445 — 10 + 24 - 293 — 24 + 5 -6 | — 2 1 um . . . — 14 — 1169 -7431 — 312 -8 I 212. Anr. I— 2V + 21D. ilii ee' 2 + 19 + 719 + 2615 + 2552 + 18 eV 2 + 6 + 49 + 3 + 4 + 5 e«' 2 y s ■ — 1 - 9 - 32 - 17 — 1 um .... + 24 + 759 + 2586 + -'5_v- + 22 llny.M. AijTiMx. Sue.. Vol. LVIL. tK mt 1 10 Y\V. KliM..->T \V. P>Rii\V.\. '/'//. ■■/''/ i if III- 213. Arg. $l' + 2iD. i. — 1 — 2 1 e' s \ eV 3 g'3 y 2 - 98 - 6 + 1 — 326 — 21 + 2 - 7 j ■■ :'3 Sum ... -103 -345 — 10 1 214. Arg. I+2F + 21D. i. 2 1 — 1 -2 -3 ey* — II -809 -45068 — 242 -309 — I | aV - 7 — 196 + 59 + 63 + 4 + 1 1 ee' 2 y 2 + 1 + 1 + 2 ! ey* Sum... + 12 - 92 + 2 -18 -992 —45100 — 179 -301 215. Arg. I— 2F + 2?D. 1 i. 3 2 1 -1 -rZ -3 ~r -55 -6331 + 39316 + 9367 + 165 + 2 «3y 2 -11 + 10 - 239 + 24 + 40 + 1 1 ee n y 2 + 10 — 22 - 4 ey 4 — 1 - 7i + 477 — 21 - 3 J Sum... — 1 —67 -6382 + 39532 + 9366 + 202 + 3 J 216. Arg. Z' + 2F + 2i"D. 1 i. , e'y 2 2 1 — 1 — 2 + 1 + 47 +392 -2195 -15 I eVy 2 1- 1 + 20 + 31 + 36 + 7 ! e'V + 2 - e'y 4 -1 - 7 + 2 + l Sum ... + 2 + 66 +416 -2155 — 7 J Motion of the Moon. 217. Arg. I'— 2? + 2/D. t. '. .1 — 1 — 2 e'y a — 1449 + 59 + 3°4 + 6 ■:. : :f eVy 2 + 4 + 24 + 83 + 5 e'V e'y 4 + 3 - 7 - 3 Sum... -1442 + 76 + 384 + 11 1 i: 218. Arg. 2^+22'jD. i 21,. 3 1 — 1 -3 -5 -7 j —2 -3 + 1 -595 - 6 + 17 + 1773 + 10 - 38 + 1228 + 3 — 10 + 57 + 3 — 1 + 1 1 i Sum... 1 —4 -584 + 1745 + 1221 + 59 I + I 1 219. Arg. l+l'+2ij). 2i,. 3 1 — 1 -3 1 -s 1 «'«'«! j <?e'y 2 a, !: : + 19 + 5 — 1 + 1244 + 41 — 20 + 143 — 2 — 4 + 230 + 7 - 4 Sum... _ , + 23 + 1265 + 137 + 233 '■':■■ :'■* 220. Ars. l—F + 2i,~D. 1 3 r ■ — 1 -3 . • " -s ee'aj | eVa, j ee'y l a x + 6 -3 — 122 - 5 + 5 — 1062 - Si + 26 -274 — 7 + 5 —4 — 1 + 2 1 Sum... + 3 — 122 — 1087 —276 -3 n8 Mr. Ernest AV. Brown, Theory of the 221. Arg. 2F + 2« 1 D. 7 2 «! + i + 254 + 584 + 258 + 1 eV«l + 3 + 6 - 4 : y i a l - 6 — 2 — 1 Sum... + 4 + 254 4-582 + 253 + 1 1 222. Arg. 4?+2z'D. i. 2 1 — 1 — 2 ~3 -4 + 2 + 72 — 2 + *953 - 5 — 10 -957 + 5 + 1 + 2 — 14 -4 Sum... + 2 + 70 + 1938 -952 + 3 — 14 —4 223. Arg. 2?+2F + 2^D. '• 2 1 -1 — 2 ~3 e 2 y 2 e 2 y 4 — 2 — 1 -105 -4005 +558 — 20 + 15 +1 + 2 — 6 — 2 —6 + 1 -3 Sum... -3 — 123 -399 6 +557 -5 -3 224. Arg. 2I— 2F + 2?T). i. 2 1 — 1 -2 -3 — 10 -45° — 1352 + 537 + 171 + 4 eV — 1 — 4 + 80 - 3 + 3 + 1 e 2 y 4 - 5 — 26 + 4 — 1 Sum . . . — 11 -459 — 1298 + 538 + 1 73 + 5 225. Arg. 4F + 2? T>. i. 1 — I ! 7 4 + 8 + 6 + 407 4- 11 + 418 + 77 _ 3 | Sum... + 14 + 74 : r "i '~ ye' 2 f yV 2 ye 4 + 5 + 8 + 2 + 633 - J 3 + 527 + 58 — 2 Motion of the Moon. 226. Al'l'. F+2/'I>. + 94476 - 876 + 24010 - 92 + 3 — "I — 121 + t — 122 — 16 + 1S5172S3 O -. 558" o o - 9 o o + iS o — I) 1 344(1 - 95a - 4857 + 588 — 9 — 2 + II + 3 1 19 -2 -2897 -19 + 3 ~ 793 -16 + 11 + 7 - S — 2 + 2 Sum... J +15 +1192 — 117262 + 18461480 -623658 -3675 -37 227. Arg. V + I+21D. 1 . i ** 3 2 1 — i - 2 -3 -4 ye + 1 + 140 + 13019 + 1014212 -167571 -6536 -80 — 1 y 3 e - 3 - 137 - 26 + 590 + 5 ye 3 + 1 + 72 + 2285 — 4001 + 357 - 66 ~I5 yee' 2 — 1 - 19 ■-• ■ ."6' + 49 + 16 yea j 2 + 1 — 1 ■iC-.:'\S:-<\?K'i ye' + 6 - 14 + 4 + 1 1 i y3 g 3 — 1 - 13 - 6 — 2 + 1 y 6 e + 3 I Sum . . . + 2 +•213 + 15122 + 1010180 — 166577 -65S0 -95 -1 ! ils 228. Arg. Y — 1+2I.T). i. 3 2 1 — I -2 -3 ye + 28 + 2600 + 201433 — 997081 -33m —401 -4 y 3 e — 1 - 41 - 997 - 3755 + 213 + 4 ye 5 + 14 + 45 2 - 884 + 1129 — 5°4 - 79 — 2 yee n - 5 73 + 7 + 44 - + 1 yea 2 + 2 — 1 ye* + 1 — 2 4- 1 + 2 + 2 — 1 y s e s - 4 + 3 + 22 - 5 + 1 r>e - ;■-. + 3 Sum ... + 42 + 3000 + 199485 -999695 -33359 -475 -6 ■ 120 ye- y 3 e' ye" yc'a^ Sum ■13 -24 Mr. Ernest ^y. Brown, Theory of the 229. Arg. F + l' + 2iT>. — 1002 + 19 - 285 — 1269 -6125 + 177 - 55° + 6 -6492 - 29443 - 16 - 253 + 23 -29689 -341 - 77 - 4 ii -3 —4 230. Arg. ¥—l f +2iT>. i. 3 2 1 — 1 —2 -3 ye' + 1 + 9 1 + 6844 + 4794 + 12073 + 89 + r y 3 e< — 2 - 5 2 - 158 + 18 — 1 yeV + 1 + 65 + 1215 + 232 + 49 + 25 + 1 ye' 3 - 6 - 5 yeV Sum ... + 2 + 154 + 8001 + 4863 + 12140 + H3 + 2 231. Arg. F 2iJ). 2!,. 3 1 — 1 -3 -5 y«, + 5 -5418 + 4810 + 320 + 1 y 3 «i + 2 + 136 - 89 + 6 ye 2 «i -37 - 74 + 73 + 26 + 2 ye'V + 1 — 1 + 1 — 2 Sum ... —29 -5357 + 4795 + 35° + 3 y 3 e 2 yV 2 Sum 232. Arg. 3F4-2z'D. -3 - 92 + 1 - 52 -143 -5978 — 10 — 311 — 6299 — 2277 + 2 + 87 + 3 -2185 -65 + 2 -63 .)/..//. n ../" the Moon. 233. Ar<j. F + 2I + 21D. 121 ■ i. 2 1 — 1 — 2 -3 -4 ye* + 20 + 1341 + 62261 — 15682 -638 -81 — 2 y 3 e 2 - 15 — 24 + 65 + 3 + 1 ye 4 + 8 + 200 - 3i9 + 54 — I ye 2 e' 2 - 3 5 — 2 Sum... + 28 + 1523 + 61913 -15565 -635 -81 — 2 ye" y :i e 2 ye 4 yeV 2 Sum.. + 1 + 1 234. Arg. F — 2^+2 iD. +53 — 1 + 8 + 2451 - 25 - 10 - 3 + 60 + 2413 -1630 — 2 + 6 + 2 — 1624 -31504 — 3*4 + 46 + 9 -31763 — 2136 + 30 — • 44 + 4 — 2146 —42 + 1 - 7 - 4 b „ii*j . I 235. Arg. F + ^ + / / + 22'D. i ■ i. 2 1 -1 — 2 -3 J yee' -4 -203 -5340 -7502 -593 -13 y 3 ec' + 4 + 6r + 22 i yeV '■ — -;2 - 40 - 49 + 14 - 7 ■ •.— 2 -. 1 yee' 3 - 3 + 3 Sum ... -6 -239 -5331 -7463 — 600 -i'5' 236. Arg. F-l~l' + 2iD. 1 i. 3 2 1 — i —2 yee' + 5 + 303 + 8975 + 5118 + 821 + 14 j y 3 ee' - 5 . — 41 ' - 27 4 '. yeV + 2 + 43 - 28 + 1 + 9 + 3 j yee' 3 - 4 + 4 : Sum ... + 7 + 341 + 8902 + 5096 + 826 + •7 ill 122 Mr. Ernest W. Brown, Theory of the 237. Arg. F + l-l' + 2iD. i. 2 1 — 1 -2 -3 yee' + 21 + 1022 + 6848 + 769 + 170 + 3 y 3 ee' - 9 — 80 + 17 yeV + 9 + 129 - 16 + 11 + 1 yee' 3 — 1 +. 4 — 2 Sum . . . + 3° + 1141 + 675 6 + 795 + 171 + 3 238. Arg. Y-I+F + 21D. i. 3 2 1 — 1 — 2 -3 yee — 1 —44 — 1302 -5707 — 1762 -50 — 1 y 3 ee' + 1 — 12 + 42 + i7 + 1 yeV - 8 — 11 + 15 - 29 — 9 yee' 3 + 2 - 5 + 1 Sum . . . — 1 -5i -i3 2 3 -5655 -1773 -58 — 1 239. Arg. ¥ + 2l' + 2iD. z, 1 — 1 -2 ye' 2 — 1 -49 -1085 -25 y'V 2 + 3 — 1 yeV 2 -15 — 10 — 10 — 4 Sum -16 -56 — 1096 — 29 240. Arg. F-2l' + 2iD. i. 2 1 — 1 ye" + 8 + 343 + 16 + 126 yV 2 - 3 — 1 + 1 yeV 2 + 5 + 47 + 4 + 9 Sum + x 3 + 387 + 19 + 136 + 1 + 1 Motion of the Moon. 241. Arg. F + I+2IJ). 1 ^ m yea i 7« 3 «i 1 '-3 + 1 ••■•-s -678 + 19 *""* 7 + 439 - 18 + 8 + 3°6 — 2 + 2 + 10 + 1 + 1 Sum ... | -7 -666 + 429 + 306 + 12 242. Arg. F-I+21J). 21,. S 3 I -1 -3 yen, y s eu, ye 3 o , — 1 — 1 — 204 + 4 - 8 + 136 + 6 O + 587 + 1 + 3 + 33 + 2 Sum ' ' ' ■ -*2 ~-2o8 + r 39 + 59i + 35 ■ 243. Arg. F + P+21J). ye'a yVa, yeV«, I Sum... 3 1 - 1 '~% + 9 + 5 + 795 — 12 + 21 + 14 — 1 + 23 + x + 2 + 14 + 804 + 13 4-26 ye'a, yVri, ye'-'c-'a. Sum. 244. Arff. F-/ / + 2? ] D. + 2 — 20 + 2 -18 Royal Astron, Soc, Vol. LVII. -788 + 10 - 28 —806 -32 -34 ■ Mm 124 Mr. Ernest W. Brown, Theory of the 245. Arg. 3F + Z+2^D. i. 1 — 1 -2 -3 y 3 e y 5 e •v*e 8 -23 - 8 - 992 - 2 - 27 -343 + 1 + 13 + 6 + 1 — 1 Sum -3i — 1021 -329 + 7 — 1 246. Arg. 3F-I+2H). j. 2 1 -1 — 2 y 3 e y 6 e y 3 e 3 -s — 2- -234 + 1 — 11 -2808 - 39 + 33 + 290 + 3 — 1 + 5 Sum -7 — 244 — 2814 + 292 + 5 247. Arg. F + 3I+21D. i. 2 1 -1 -2 3 - 4 ye 3 i 7 3e3 j ye 5 + 2 + 1 + 124 — 2 + 17 + 4015 - 4 - 27 -1528 + 6 + 6 + 8 + 1 -7 — T i Sum 1 + 3 + 139 + 39 8 4 -1516 + 9 -7 — I 248. Arg.F-3Z+2*D. i 4 3 2 1 -I -2 ye 3 y 3 e 3 ye 5 + 1 + 32 — 1 + 22 — 1 + 2 S3 + 2 -1570 - 19 + 4 — 146 + 3 - 4 -4 Sum ... + 1 + 31 + 21 + 255 -1585, -147 -4 Motion of the Moon. 249. Arg. 21D. <^5 j 3 2 1 1 I + 11 + 1568 + 245748 + 34226987 e 2 + 16 + 98S + 37988 e' 2 - S — 232 + 11 ¥■ — 12 - 9 2 4 a * 2 + 4 e 4 + 5 + 86 - 40 + 1 eV 2 - 5 25 <?V — 10 - 187 + 3 e'y 7 1 + 1 — 1 Sum . . + 32 + 2607 + 282333 + 34227001 250. Arg. I+2H). i. 3 2 1 -1 — 2 3 -4 I e + 3 + 3°5 + 27534 + 1866057 + 345043 + 5396 + 61 +1 ! e 3 + 3 + 126 + 3493 681 - 128 + 678 + 25 +1 ee' 2 • ■ - : '— ; - 2 - 39 - 13 - 118 — 11 :..ey*. ;■ 3 — 104 + 3 2 - 1687 - 47 — 2 ea^ + 1 + 4 e s + 1 + 10 - 6 — 1 + 2 1 «V - 3 - 18 — 10 — 7 Sum... + 13 + 3 + 7 + 433 + 30861 + 1865398 + 343 rl 7 + 6008 + 86 +2 j 251. Arg. /' + 2/D. *• 1 e ' 2 t '";■■— t. 2 — 3 -33 -2569 — 3924 + I74I5 + 223 + 2 1 I e*e> — 20 - 45i — 127 + 1874 + 120 + 4 e' 3 + 2 — 14 — 2 i e y + 17 + 47 - 73 — 2 eV — i j Sum ... -53 —3004 — 4002 + 19202 + 339 + 6 126 Mr. Ernest W. Bkown, Theory of the 252. Arg. 2? 1 D. e' 2 a l y 2 a, Sum + 1 + 1 + 81 -62 + 1 + 3 + 23 1 — 9800 — 192 — 2 + 242 -975 2 253. Arg. 2 1 + 21D. ?'. 3 2 1 - 1 -2 -3 -4 e 2 + 4i + 2546 + 101788 -3052 + 3760 + 98 + 1 e 4 + 1 + 13 + 299 - 125 + I + 4 + 11 + 1 eV 2 — j - 9 - 5 e 2 y 2 - 9 + 3 + 12 - 37 Sum... + 1 + 54 + 2833 + 101657 -3039 + 3722 + 109 + 2 ee'y 2 Sum... _ 4 — 426 - 61 + 2 -485 254. Arg. l + l' + 2iD. -9536 - 56 - 7 + 97 -9502 + i45 I 5 + 7 - 5 - 62 — 2 "~ 3 + 619 + 11 + 63 + 4 | — 2 - 6 1 + 14455 +674 +15 255. Arg. I-V + 21D. i. 3 2 1 - 1 - 2 •-3 ee' + 1 + 45 •f 2122 + 11654 — 2214 - 90 -3 ■■ eh' + 15 + 195 + S — 24 — 12 — 2 ee' 3 - 4 4- 7 + 4 ee'y 2 + 1 - 8 — 124 - 26 Sum... + 60 + 2305 + 1 1542 — 2260 — 102 -5 '.. y- ■Sum ... 256. Al'g. 2/' + 2/'D. I — 1 -2 — 2 -26 -S 4 - 4 + 2 + 853 + 7i - 4 + 19 + 9 -28 -86 + 920 + 28 12; 257. Arg. 2F + 2&T). i. 1 -1 -2 -3 + 1 7 2 + 9 + 667 — 1 07 1 + 41 eV -17 -788 + 16 - 9 e'V + 2 V 4 — 1. + 1 — 1 um ... - 9 — 124 — 1052 + 3i + 1 258. Arg. I+21J). 2!",. 3 1 -.1 -3 eaj + 2 — 1 103 + 120 -378 e 3 a t -7 — 16 + 2 - 15 e-Ai + 2 + 29 — 4 + 8 Sum... -3 — 1090 + T18 -3*5 -3 ■259. Arg. I' + iiJ). 2J. 3 1 — i ~~ .' e'a, 1 + 20 + 1464 -40 + 10 e 2 e'a. + 7 + 5° - 3 1 «V«i 22 + 3 >Sum ... + 27 + 1492 -37 + 7 | 128 Mr. Ernest W. Bbown, Theory of the 260. Arg. 3I+21T). i. 2 1 — 1 -2 -3 -4 e* + 5 + 219 + 6231 — 1192 + 76 + 47 + 2 e-' + 2 + 24 - is + 1 + 1 e 3 y 2 — 1 + 4 — 2 — 2 Sum ... + 7 + 243 + 6215 -1187 + 74 + 46 + 2 261. Arg. Z+2F + 2z'D. ! i. 1 -1 -2 -3 ey* + 1 + 63 -847 + 15 + 2 e 3 y 2 -73 + 11 — 1 ey 4 + 3 . Sum... + 1 — 10 -833 + 14 + 2 262. Arff. /— 2F + 2/D. i. 2 1 — 1 — 2 — 1 ey 2 — 2 -476 — 7063 - 88 <3 3 y s -3 — 1 + 29 — 24 ey 4 _ 4 — 102 — 1 Sum ... -s -481 -7136 — 112 m Mill,,,, „f //,(■ .]/, ,.,!,. 120 Section (v). The Final Values of the Coefficients in Longitude, Latitude, and Parallax. 263. The following tables, giving the final values of the coefficients, are arranged, first, according to the order and composition of the principal characteristics ; second according to the signs of the multiples of I, I', F : and third, according to multiples of 1) in descending order. In the first column, headed " P. C," is given the principal characteristic. In the second, third, fourth, and fifth columns, headed " /." " /'," "J>\" « ]) » are O .[ von t ] ie multiples of those arguments (Dklauxay's notation) to which the coefficients in the last column correspond. The characteristic is understood to belong to all coefficients down to the next printed characteristic ; a similar remark applies to the multiples of /, If, F. As stated earlier, the system of axes used is the same as that in which Delauxay expressed his final results. These coefficients are all definitive for the corresponding arguments, with the following exceptions : — (a) Small changes due to possible changes in the values of the arbitrary constants. (/3) Small additions due to the terms arising from the perturbations noted in (a), {!>), ((■) of § 4, Chap. I. These are very minute and are easily obtained. They are, however, more simply treated by the method of the variation of arbitrary constants^ and will therefore be given with the treatment of the planetary inequalities. The results have been discussed, and a comparison has been made with those of Hansen elsewhere.* * E. W. Brown, " The Final Values of the Coefficients in the New Lunar Theory," Monthly Notices, January 1905. HO Mr. Ernest W. Brown, Theory of the 264. Longitude. Coefficients of Sines. P.O. J I. 1'. F. D. ; Coeff. I OOO 8 : + O'OOI 6 + -127 i 4 + i3"9° 2 1 2 + 2369-899 e 1 6 + -023 4 + 1-979 2 + i9 r 954 + 22639-580 — 2 - 4586-438 -4 - 38428 -6 - - 393 -8 •004 e' 010 6 — '004 4 - -289 2 - 24-451 — 668-944 — 2 - 165-351 —4 1-879 -6 — '024 "i 0005 + -004 3 + -402 1 - 124-785 e- 200 6 + '004 4 + -213 2 + 14-387 + 769-021 — 2 — 211-658 —4 - 3°-773 -6 '570 -8 — -009 ee' 1 1 4 - -051 2 | - 2-925 j — 109-804 ! -2 — 206-219 —4 - 4-396 -6 — -072 -8 1 — -ooi P.O. | i /. V. F. D. Coeff. ee' | 1 —1 6 + 0-005 I 4 + -283 { 2 + 14-595 + I47-878 — 2 + 28-511 -4 + -637 -6 + -on e' 2 0204 — -003 2 - -189 - 7-505 — 2 - 8-n6 -4 - -151 -6 — "002 7 2 0026 — 'OOI 4 - -085 2 - 5-741 —411-614 — 2 - 55-174 -4 + -025 -6 + -ooi e'cq 1003 — '002 1 - 8-441 — 1 + 18-554 -3 + 3-206 -5 + -014 e'aj 0105 + -002 1 3 + 'IS 1 + I7'992 — 1 + -559 -3 - -066 -5 — "OOI .1/.. '!••!,- ■■/' '''". M"->ii. 131 Longitude. Coefficients of Sines (continued). : P.O. j I. 1'. P. D. r.vff. . e 3 1 ; 3 4 + 0'02I ■ 2 + ro6o \- + 36-124 i — 2 -I3T93 : fe': : ';v>:0>.v : o> —4 — 1-187 :'l?Sxo:' ;: ' :: ^v:^ -6 - '293 -8 — '009 : eV 210 4 — -007 ' 2 — '290 1 - 7'6S9 — 2 - 8-638 §>ft?SSS-:X? —4 - 2-743 -6 — -091 -8 - -003 2 —1 4 + "033 2 + 1-182 + 9*7*5 i — 2 - 2-497 | -4 + '360 i -6 + "014 ! ee' 2 1202 — '014 i — 1-169 ; — 2 - 7'43i -4 - "3 "2 1 j -6 — -008 i 1 —2 4 + -024 2 + '759 ! + 2-586 < — 2 + 2"539 -4 + '022 c '3 0300 - t°3 — 2 - -345 | 1 -4 — "010 P. c. "'. P. D. Coeff. ■ -, ' '■V e 2 a! 024 2 o — 2 -4 —2 6 4 2 o — 2 -4 -6 1 24 2 o — 2 -4 1—2 2 o — 2 — 4 003 1 — 1 -3 -5 -7 1 o 3 1 — 1 -3 -5 — C018 — "992 — .\y\:z — -i79 — -301 — -ooi — -067 - 6-382 +39-532 + 9-366 . + '202 + + + •O03 •0G6 + '416 - 2-iS5 - '007 - 1-442 + + + •076 ■3M •on — -004 - -584 + 1745 + I-22I + '°59 + 'OOI + -023 + 1*265 + -137 + -233 + 'OOI llll mm mmm 111111 8 Royal Astron. Soc, Vol. LV1I. 132 Mr. Ernest W, Brown, Theory of the Longitude. Coefficients of Sines (continued). P.O. I. 1'. F. D. 1 Coeff. ee'a x i —i o 3 + 0'003 i — '122 — i -1-087 -3 — -276 -5 - "003 e' 2 n. O 2 ° 3 — '002 i - '°39 — i — -042 -3 — - oo6 7 2 «i o o 2 3 + '004 i + '254 — i + -582 -3 + '253 -5 + - ooi ; e* 4 o o 4 + "002 2 + '070 O + 1-938 — 2 - - 95 2 — 4 + -003 -6 — "014 -8 — "004 I eV 3 i O 2 — '° 2 S O - - 552 1 — 2 - -483 1 — 4 — TOO j -6 - - °39 -8 — "OOI 3 -i o 4 + '°°3 2 + -088 + -682 — 2 - -183 -4 — "029 -6 + '005 P.O. e'e ■ F. D. eV 1 -3 o o — 2 -4 -6 o 4 2 o — 2 —4 -6 o o — 2 -4 o 4 2 o — 2 o o — 2 2 4 2 O — 2 -4 -6 -2 4 2 — 2 -4 -6 Coeff. — C067 — •298 - •161 — •008 + •003 + •062 + •197 + •255 + ■036 + •001 — •018 — •250 - •016 + •001 + •032 + •051 + •003 — ■001 — •013 — •003 — •123 ! -3 - 996 + •557 ! - •005 j — •003 — •on — '459 — ] [•298 + •538 + •173 + •005 T Million hi ihr Mr.,,, i. '33 Longitude. Coefficients of Sines {continued). : P.C. J. J'. F. D. Coeff. ee'y 2 I I 2 2 + 0'0I2 o + "263 — 2 + -059 . -4 — '024 -6 — 'ooi I I —2 4 4" '002 2 + -083 o - -083 — 2 + '4-'7 -4 + -019 I —I 2 4 — -002 2 — '064 - '3°4 -2 4- '002 -4 4- -018 I —I —2 4 — '00 7 2 - - 372 ■ o + -083 — 2 - -065 j -4 — - 002 | e'V O 2 2 O + '004 — 2 — "066 " •• -4 — - 002 O 2 —2 2 — '025 — - 002 — 2 4- -016 V 4 4 2 4- '014 4- -418 — 2 + "074 e 3 a t 3 1 — "042 — I + "130 -3 + -045 -s 4- - oi6 -7 + "OOI P C. I. I'. F. D. Coeff. eVa ! 2 1 P 3 1 — 1 ~3 -5 4-o - oo3 4- -092 4- - oo6 + "084 4- '006 2 — 1 1 — 1 -3 -5 — "014 — '3S 2 + "042 — -003 ee n a x I 2 1 — 1 -3 — -008 — 'O02 -»• "012 I — 2 -1 -3 + - o°3 4- "ooi e' 3 a l O 3 ■ I — x — 'OOI — '002 ey 2 oi I 2 1 -— 1 -3 -5 + -045 + -024 + -030 4- "002 1 -2 3 1 — 1 -3 — - OIO — '041 — '016 — "OTI e'y^a s ; • ' i ,' 2 3 1 — 1 -3 — "OOI - "035 + - oi3 + '020 :. o : 1 -2 • 3 1 — 1 4- '009 — 'OOI — - 002 m 134 Mr. Ernest W. Brown, Theory of the Longitude. Coefficients of Sines (continued). P.O. i. I'. F. D. Coeff. e'° 5 o O 2 O — 2 -4 + o'oo5 + 'US — '069 + '004 eV i 4 i O 2 O — 2 -4 -6 -8 — '002 — -040 — -030 '000 — "002 — 'OOI ! 4 — i O 2 o — 2 — 4 + -007 + -048 — '019 — - 00I •J /o e 6 e " 3 2 o o — 2 — 4 -6 — -003 — "016 — -006 — '003 3 — 2 O 2 O — 2 -4 -6 + '°°5 + '016 + - OII + '004 4- '001 eV 3 2 3 o o — 2 —4 -6 — 'oo r — '010 — -008 — 'OOI 2 -3 O 2 — 2 + '°°3 + '004 + 'OOI "\ 9 & y 3 o 2 4 2 O — 2 — -003 — 'Oil - - 33° + -092 P. C. I. I'. F. D. Coeff. 3 —2 4 — cool ! 2 - . '°33 | - -055 — 2 — '005 —4 + -009 -6 + "003 e'Vy 2 2 122 + '002 + - °43 — 2 + -028 2 1 —2 2 + '009 4- "026 — 2 4" '022 -4 + '016 -6 + 'OOI 2 — 122 — '009 - '053 — 2 + "004 2 — 1—2 4 — 'OOI 2 — '029 — '024 — 2 •000 -4 — '002 ee'V 1 220 + "003 — 2 + '004 -4 — 'OOI 1 2 —2 2 — '002 'OOO — 2 + -015 -4 + 'OOI 1 — 222 — '003 — -005 — 2 + -007 -4 — 'OOI \rnr-. IB M>Al"H <■/ !/», M<vi< 135 Longitude. Coefficients of Sines (concluded). I- i I' '••'•• ■/. .•• .vl F. D. Coeff. r. c. I. V. F. D. Coeff. t .,,'i y l 1 — 2 — 2 4 — O'OOI e i y-a l &M-- 0-2 3 — - OOI 2 — - oi6 1 — 'OOI •000 — I + 'OOI ?— 2 — '005 -3 — '003 e '3 y 2 ■3 2 —2 — - 002 ee'y' 2 a x I 1 ■ 2 1 — "006 3 — 2 —2 + 'OOI — 1 + 'OOI ey 4 1 4 2 + '003 -3 + "002 + -090 I 1 —2 1 — "002 — 2 + '009 — 1 *000 1 -4 4 — "OOI -3 — - OOI 2 + "OOI I -1 2 -3 — - 00I - -080 I -1 —2 3 — 'OOI — 2 — '019 1 — -004 e'v 4 1 4 — 2 — 'OOI + *oo3 — 1 -3 •000 + 'OOI 1 — 4 2 + '002 •000 y*«i 04 1 — 1 — "OOI — "OOI — 2 — "OOI e° 6 000 + -007 e 4 a! > 4 1 — 1 -3 -s — -003 + *OIO + "002 + "OOI «v 4 022 — 2 -4 — "OOI — -025 + 'OIO — -ooi | i 1 3 1 1 — 1 -3.' + -007 — 'OOI + -003 4 —2 2 — 2 — "OOI — -007 + - 002 -5 -f- - 002 ey 2 042 + "OOI 3 — 1 1 — *002 + -on ] — 1 - -023 2 —4 + "OOI | -3 + "007 — 2 - -003 I e 2 y 2 ai 2 2 1 — 1 + - oo6 — -003 -4 — '001 ! 111 (11 Ijjjjjgl 111 mm jjijjj mBSm mm t|jj ilBl 1§ m Wi m m m I illP JUl 136 Mr. Ernest W. Brown, Theory of the 265. Latitude. Coefficients of Sines. P.O. I. V. F. D. Coeff. r 0016 + 0-015 4 + 1-192 2 + 117-262 + 18461-480 — 2 - 623-658 -4 3-675 -6 - -037 ye 1 1 6 + -002 4 + -213 | 2 4- 15-122 + 1010-180 -2 - 166-577 -4 6-580 -6 •095 -8 — -ooi — i 1 6 + -042 4 + 3-000 2 + 199-485 - 999-695 — 2 - 33-359 —4 - '475 -6 — -006 ye' 1 1 4 — -024 2 — 1-269 — 6-492 — 2 — 29-689 —4 — -418 -6 — -006 —1 16 + *002 4 + -154 2 + 8-001 + 4-863 — 2 + 12-140 -4 + -H3 -6 4- '002 P.O. 1. v. F. D. Coeff. y«i 1 3 1 — 1 -3 -5 — 0-029 ~ 5-357 + 4-795 + -35° + "003 y 3 3 4 2 — 2 — -003 - -143 — 6-299 - 2-185 ye' yee -4 -6 2014 2 o — 2 —4 -6 -8 -2018 6 4 2 o — 2 —4 -6 1 1 1 4 2 o — 2 -4 -6 — -063 — -ooi + -028 + 1-523 + 61-913 -I5-565 — -635 — -081 — -002 + "OOI + -060 + 2-413 — 1-624 -31763 — 2-146 — -048 — -ooi — -006 - -239 - 5-331 - 7-463 — -6oo — -oi5 ^m. M 111 Mi'ft'on of tin Moan. i.S7 Latitude. Coefficients of Sines (continued). m ■p 1 m m I I p. ('. I. 1'. F. D. Coeff. j y ee< ■ — i — i i 6 +0-007 j 4 + "34i 2 + 8 - 9<D2 O +5-096 | — 2 + -826 J -4 + -017 i —i i 4 + '030 2 + 1-141 O + 6756 W''-i'-% • —2 + 79S %::■ :%B.. —4 + -171 f ; ^S:'¥. -6 + '003 : H-' .!'•;:;';-.'" :"; — i i i 6 — 'OOI 4 - -051 u:;^' : W:- ■ 2 -i-323 1 O -S-6SS —'2 -1-773 -4 - "OSS -6 — -ooi ye' 2 2 1 2 — - oi 6 1 - -°5 6 — 2 — 1 '096 i -4 — '029 . O —2 I 4 + -013 K : :': : ; : : /^: 2 + -387 + -019 — 2 + -136 -4 + 'OOI yea. I o I 3 — -007 1 - -666 — i + -429 -3 + -306 -5 ! + -012 1 P. C. I. 1'. F. D. foeff. yea, -1 1 5 — - 002 ■■■■■:. 3 — -208 I + -I 39 1 + '59' -3 + '035 ye'a, 11 3 + -014 1 + "804 — 1 + -013 -3 +■■ '026 —1 1 1 - -018 — 1 - -806 -3 - -°34 y 3 e 1 032 - "031 — 1 "02 1 ! — 2 - -329 -4 + '007 -6 — -ooi -1034 — -007 2 l;: i:; .'; ? ?li<|: — 2-814 — 2. + -292 -4 + -005 yV 01 3 a + -on — 2 - -°93 —4 — -006 —1 3 2 1 — -007 + -ooi — 2 + -056 1 -4 | + -003 ■ jjj tlBl i38 Mr. Ernest W. Brown, Theory of the Latitude. Coefiicients of Sines {continued). P. C. | 1. V. F. D. Coeff. P. C. I. 1'. F. D. Coeff. ye 3 3014 + 0-003 yeV — 2 1 1 6 — O'OOI 2 + '139 4 — '029 + 3'9 8 4 2 + '056 — 2 -1-516 - '303 -4 + -009 — 2 — -129 -6 — -007 -4 — -005 -8 — 'OOI yee' 2 1 2 1 - '°55 —3 01 8 + -ooi — 2 — -272 6 + '031 —4 - '°34 4 + "021 -6 — -ooi 2 + '255 — 1 —2 1 6 + -ooi -1-585 4 -f -02 2 — 2 - -i47 2 + '319 —4 — -004 + -062 yeV 2 1 1 2 — -027 — 2 + - oo6 — -644 1 —2 1 4 + *002 — 2 - -657 2 + -054 -4 - '053 + -117 -6 — -on — 2 + -107 — 2—1 1 6 + -009 -4 + -004 4 + - 2I7 — 1 2 1 4 — -ooi 1 2 - -063 2 - -115 ! + -314 — -096 _ *> + -063 — 2 — -069 -4 + '001 -4 — '003 2 -- 1 1 4 + '003 ye' 3 3 1 —2 - '037 2 + -114 -4 — - 00I + "809 -3 1 2 + -014 H_ O - -084 7 3 «i 3 1 + -006 -4 + '002 — 1 + '032 -6 + '002 -3 + -oio \f, ,//',, n >>/' tin MdiO) \Vj Latitude. Coefficients of Sines (continued). P. c. I. V. F. D. Coeff. j 2 6 i 3 — i -3 : -.5 : . — o-ooi 1 - -065 + "112 + '039 + "oos p:"'' : - : 1 — 2 Q i S 3 , . i — i -3 - -oos | - -049 - -078 + -036 + '003 , y«e'a , I I i 3 i — i —3 -s + '002 + •101 — -on + '021 + 'OOI — I — I i 3 i .' — * -3 •/•- "OI3 ■:' + 'OOI - '034 — "004 i | I —'I i i — i ~3 -s - -009 — *oo6 1 - -013 — 'OOI 1 V- • ' — I I ' 3 I — I -3 + '022 - "°S6 + "020 1 + '002 [ yc n , H O 2 i i — i -3 — \002 I — '002 + 'OOI O — 2 i i ■ + '002 1 — i i + '002 ! P.O. 1. V. F. D. Coeff. r' S + o'oo5 + - 002 •y 3 e 2 2 3 2 — 2 .—4 • — '004 — 'ii6 — '022 + "005 I I- : ■.■■■'.' 1 | ' . — 2 ' ■ : 3 4 ' .2 — 2 ;-4 — - oo6 — - o66 + -130 + 'OTO + 'OOI i -fee' 1 •■ I. 3 —2 + -007 — 'Oil — 1 — $.-; 3 2 • ~2 — -on + '008 1 — t 3 2 __2 — '002 — 'O06 + 'OO3 [■ : — 1 I 3 2 — 2 + *002 — '009 + '017 i 7V 2 O ; 2 3 - 2 -4 - '003 — 'OOI O — 2 3 - 2 + 'OOI ye* 4 I 2 . 2 -4 ; + 'Oil + -266 - -135 + '007 -4 O 1 6 .■ ..... •... ... ■... ,-.■..; 4 i 2 + 'OOI •000 + -025 ; — -091 . —2 • — '010 1 Royal Astjkon. Soc. Vol. LVII. .»:> Wmk0&£Wyix ^pllllll ^wafS?W::5a ■Ills: ^HsR?;?;::lsss J||||p:f:-:©; : :WK : S ^^^:*;#:%>> : >S: : ::: 140 Mr. Ernest W. Brown, Theory of 1 he Latitude, Coefficients of Sines (continued). P.O. 2. V. F. D. Ooeffi yflV 3 1 1 2 n — o'oo3 - -063 • .'. ~ z : - -056 -4 + 'OOI -6 — 'OOI |;; : i;iffii-i:±S; : ffi:':| -3 - 1 1 6 + -004 4 + '002 2 #■ ;OIO + '024 — 2 + -005 3 -1 1 2 ''.•;+.•'; 'cJii, + -076 — 2 — '019 -3 1 1 6 — 'OOI 4 + '002 2 + '002 — "024 — 2 — 'OIO yeV 2 2 2 1 — 'QOS —2 — '023 —4 — -003 -6 — "OOI — 2 —2 1 6 + 'OOI 4 + '013 2 — - 002 ! + - °03 , I 2 — 2 1 2 + "007 I + -016 I — 2 + "013 -4 + "OOI ' — 2 2 1 4 — -003 1 2 : -^ '006 — •006 — 2 — '005 i P.O. 1 '■" ■ ' I. I'. F D Ooeff, j ; yee' 3 f 3 1 -r-2 -4 — O'OOI i . : -'.-I — -oio — 'OOI — I -3 1 4 + 'OOI 1 -f "OIT + 'OOI ] :-:^ :::y I -3 I + '002 + 'O02 , — I 3 I 0; — 2 : — '002 i — '002 ] y'eaj I 3 * 1 — 1 -3 -5 + '002 + - O04 1 + '002 I — 'OOI : —I 3 3 1 ■ — i . -3 + 'OOI + "003 — "OOI — "OOI yVaj 1 3 1 — 1 _3 — 'OOI •000 + 'OOI — i 3 — 1 — 'OOI ye'ai 3 1 1 — 1 -3 — '006 1 + '°'3 + '003 : ; yy : ; : ■ : ; -3 1 5 3 1 — 1 — -002 •OOO 1 — 'ooi; '• 1 + '002 yeVtq '"2 I 1 1 — 1 -3 + 'OIO — 'OOI + '°°3 I ■ in m m m !;• Mullmt uf tin. Moi'U. Latitude. Coefficients of Sines (concluded). Ml i m I 1 P.O. I. V. P. i>. Coeff, ■ye i e'a l —■?. — I I 3 i — C003 — '002 — -003 2 — I X i — i -3 — "O02 — 'o[6 + 'OOI — 2 ; I I 3 i - "°°3 + 'oi7 + "OOI y 6 e I 5 o — 2 + -002 + 'OOI — I o 5 O — 2 + '003 + "OOI P. c. , v. F. n. Coeff. 1 y'V o 32 o — O'OOI — '014 -3 3 4 2 O — 2 — -003 + '002 + 'OOI + 'OOI ye 5 5 I 2 O — 2 -4 + 'OOI + -oi8 — '012 + 'OOI -5 o I 2 O — 2 + '002 — - oo6 — 'OOI jjilfll ■3 mm WM llll Wm '4BBm mm llll I 1 mm III! *»&i ■ ■r 14^ Mr. Ernest W. Brown, Theory of the I: 266. Parallax. Coefficients of Cosines. P. c. I. V. V. D. 1 Coeff. P, C. I. 1'. F. D. . Coeff. I 000 6 + 0-0032 ee' 1104 — 0-0012 4 + "2607 2 j - -0485 2 + 28-2333 — ^s 02 + 34227000 — 2 + 1 '445 5 e 100 6 + '0007 —4 + -0674 4 + '0433 1 -6 + '0015 2 + 3'°86i i 1 —1 6 + "OOOI + 186-5398 4 4- -0060 — 2 + 34'3"7 2 + '2305 -4 + -6008 + 1-1542 -6 + -0086 — 2 — -2260 -8 + "0002 —4 — -0I02 | e< 010 4 - -0053 -6 — -0005 2 — '3OO4 e n 0202 — -O028 ! — "4002 - -0086 1 — 2 + I-9202 — 2 + -0920 1 — 4 + '°339 — 4 + -0028 ' -6 + -0006 7 2 0022 — -0009 a, 000 5 + -oooi — '0124 ; 3 + '0023 — 2 — -1052 1 - '9752 — 4 + -0031 e 2 200 6 + "OOOI -6 + 'OOOI 4 + -0054 ea { I 1003 — -0003 2 + "2833 1 — -1090 + 10-1657 j — 1 | + -0118 — 2 - '3°39 1 -3 - '0385 — 4 + '3722 I -5 - -0003 -6 + -0109 e'a, | 0103 + -0027 -8 + -0002 I i i [ — 1 ! + "1492 — '0037 1 -3 | + -0007 Motion of the Moon. *43 ■mi i Parallax. Coefficients o£ 'Cosines (continued). ■P • .* , Mi P.O. /. Z'. F. D. Coeff. e 3 3004 1 + ©•0007 2 + -0243 O + '6215 — 2 — '1187 - 4 + -0074 -6 + -0046 -8 + "0002 eV 2 104 — ■000 1 2 — -0051 - '1039 — 2 — '0192 -4 + '0324 -6 . + '0017 2 —1 4 + "0007 2 + -0213 + "1270 — 2 — '0017 —4 - '0043 -6 — '0002 ee' 2 1202 + '0001 — 'oio6 — 2 + '0485 -4 + *oo44 -6 + '0002 1 —2 4 + -0005 2 + '0112 + -0196 — 2 — "0213 —4 — '0003 e' 3 0300 — '0002 — 2 4- "0036 —4 + '0002 P.O. I'. F. ey' ey z y 2 a, 022 o — 2 -4 -6 —2 4 2 o — 2 -4 12 — 2 -4 1—2 4 2. o — 2. 001 — 1 -3 . -S 1 o 3 1 — 1 —3 1 o 1 — 1 . — 3 2 o 1 ' — r ■ o 2 I — 1 -3 Coeff. 4-o'oobi — 'ooio — -0833 + '0014 + '0002 — -0005 — '0481 — -7136 — 'Oil 2 — '0001 +' "00 1 3 — - oo66 + '0005 — 'OOOI + '0014 + "0017 +• "OOOI — 'OIOO + -0155 — -0088 — - ooo8 + -0003 + "0164 •0000 — -0025 — "0014 •0000 + -0036 — -0003 + '0003 + 'OOOI :+ 'OOOI + "0071 — '0017 - : - : **' r iiis§ •:p sill ' ; i '44 Mr. Ernest 'W. Brown, Theory of the Parallax. Coefficients of Cosines -(continued). P. 0. I. 1'. F. I). 1 i Goeff. 1 ; |:;:; : ;;;x;:::;::;:;:;;:;:; ; ; : ;::;:-; : ;xi::: } P.O. I. 1'. F. D. [ Ooeff. e* I 14004 -| + 0'OOOI g2y2 2 . ' 2. z '• " — O'OOOI 2 + - ooi8 1 + '0004 + -0401 1 ■' • • ; '.. —2. — '0090 ■'..'— 2 ' •. - -0x30 t' -4 + '0002 -4 + 'OOOI 1 2 — 2 2 — "0053 -6 + 'OO02 1 + '0004 i -8 + "OOOI I' • ■ . — 3 • — '0141 eV 3102 — "0006 1 ■'■•• ' -4 — '0004 — '0097 ee'y % 1 1 2 + 'OOOI ™-:2' - '°°45 — 2 — "0032 —4 + "0006 | _ 4 + 'OOOI -6 + "0005 1 1 — 2 ' 2'. + '0006 -8 + 'OOOI ; ° ■ + '0024 3—10 4 + "OOOI — 2 — '0006 2 -f -0017 1 —1 : 2" ■ * : 2 — 'OOOI 4- '0115 O + '0003 — 2 — -0017 .'' —2 + '0004 -4 + '0002 1 —1 ~2 4 — 'OOOI -6 — "OOOI 2 — -0027 eV 2 2200 — '0009 ° ! — '0029 . '■■-rz .-•'• — -0009 e'iyi 2 2 — 2 — -0004 —'4 + "002O y* 4 ' •0000 -6 + ; •OOOI e 3 a l 3 1 — '0009 2 — 2 4 + 'OOOI —1 + '0017 2 + "0013 1 -3 + "OOOI + '0024 —5 ! — '00O2 — 2 — 'OOOI e i e'a l 2 1 3 1 + ;bbp2 -4 — -0005 1 + '0015 ee' s 1300 —2 —4 — '0002 + '0014 + '0002 — I — '0O02 — 3 | — '0005 — 5 i — '0002 j 1-30 a + '0004 2 "—1. 1 I •'. "^- 'PP°5 ■ ..0 :•'.: + '0004 — 1 | — '0028 1 • .~* 2 + "0002 -3 — '0005 -+■ -0002 m m m m m Si ■ m pi mmlmm i$ m is. Wm Motion of the Moon. 145 Parallax. Coefficients of Cosines (concluded). P.C. I. V. F, I). Coeff. t I • 2 . 1 : — I- -3 -5 + 0"Q002 4- "ooio + "0002 . '1 — "O0O2 ! I — 2 3 1 — 1 -3 — ' "0002 •OOOO + '0006 + "0004 e'y i a l 1 2 — 1 + 'OOOI 1 I — "000 T 1 — 2 3 : + "OOOI 1 : : : i — -0003 :1 P.C. I. V. F. D. Coeff. e* 5 2 , : ~2 : — 4 + 0"0002 + "0026 — "0012 + "OOOI e 3 y 2 3 2 2 O — 2 —4 — 'OOOI "OOOO — "OO09 + "OOOI 3 — 2 2 ; —2 -4 — -0005 — -0003 + 'OOOI — "0008 «y 4 1 0^ —4 ' • • 2 ■ 0. . + "0002 — "OOOI Ha.vcrford College. : 1 904 December 3. Errata (additional to those given on p. 202 in Part II.) Part I. (vol. liii.), p. 43, line 12, for "equal masses " read " masses equal to their actual masses." „ „ 46, „ 18, for "parallactic inequality" read "the principal parallactic inequality in longitude." si si 83, „ 2 (in some copies), for " v r _ c , _ ( _ l " read " i',_,i j-w-" Part II. (vol. liii.), p. 166, line 21, for the denominator " «/%' " read " u'^." Part III. (vol. liv.), p. 5, line 6 from bottom, for " K % " read " Il x ." „ ,, 19, „ 6 „ for "+"ooooi o" read " ■—••ooooo 9." ,. ,. „ 4 „ for " —"00019 2 " rea d " —'00012 9." „ „ last line, for " +-19822 " read " +-19828." Theory of the Motion of the Moon; containing a New Calculation of the Expressions for the Coordinates of the Moon in Terms of the Time. By Ernest W. Brown, M.A., Se.D., F.R.S. [Received April 13; read May 8, 1908.] PART V. CHAPTERS X.-XV. The previous parts of this Memoir have been published in the Memoirs of the Royal Astronomical Society under the same title in 1897, 1899, 1900, 1905. They con- tained the solution of what I have called the main problem — the motion of the Moon under the attraction of the Earth and the Sun. which are supposed to move round one another in a fixed elliptic orbit in the plane of reference, the three bodies being treated as spheres. This fifth and last part treats of the effects of all other gravitational causes — the direct and indirect attractions of the planets, the deviations of the masses of the Earth and Moon from mechanical sphericity, and various minor perturbations which had been specifically excluded. The problem presented by these additional perturbing forces is a complicated one. In order to clear the ground, it was first assumed that each force contained a small factor whose square could be neglected, so that the perturbations could be separately considered ; then the perturbations were supposed to be expressible by a series of secular and periodic terms, each of which, under the same assumption, could also be separately considered. Chap. X. therefore contains the method for finding the effect of a perturbation expressed by a periodic or secular term, and in Chaps. XL, XII, XIII. this method is successively applied to direct and to indirect planetary action and to the action of the figures of the Earth and Moon. In Chap. XIV., in addition to minor perturbations, the effects of including the squares and higher powers of the small factor is considered : a few very small terms were found. In Chap. XV. the results from all these perturbations are gathered together, so that the final expressions for the coordinates of the Moon in terms of the time are obtained by adding the results in Chap. XV. to those previously given in Chap. IX. EoYAL ASTRON. SoC, VOL. LIX. I p^^^^^^fp^^^^l^^^^p^lplp^^^p^^pppl 2 Mr. Ernest W. Brown, Theory of the The whole question of these perturbations, in spite of this division into parts, is still complex, and, in preparing the results for publication, the choice of the portions to be presented in detail was an embarrassing one. It was made more so by the conditions which rendered necessary a previous publication of the work for the direct planetary inequalities in separate form.* This work included a full investigation of the equations of variations which are needed for inequalities arising from all sources, and there was thus a choice between repeating this investigation and leaving the present Memoir incomplete. The same difficulty arose with the direct inequalities and with some other subjects which I have discussed in previous papers. I finally adopted the plan of inserting this previous work either when it was essential for clear presentation, or when it was sufficiently brief to occupy but little space, or when the proofs could be considerably improved. Thus, in Chap. X., the equations of variations, the idea of which is due to G. W. Hill, are rapidly put into the required form (A.P.E., Sect. I.); the formulas for obtaining derivatives with respect to n from a theory in which the numerical value of n'jn has been substituted [Trans. Amer. Math. Soc, vol. iv. ) are deduced in a few lines; the methods for dealing immediately with non-periodic terms {Proc Lond. Math. Soc, vol. xxviii., and Trans. Amer. Math. Soc, vol. v.) are partly developed, but the results (ib., and Monthly Notices, vol. Ivii.) are only quoted. In Chap. XL the proof of the theorem on which the method for the direct inequalities is based (A.P.E., Sect. II.) is exhibited in a simple form, but the full algebraical results are merely quoted ; for the rest of this part of the subject the methods are described in general terms and the final results alone given. In Chap. XII. a theorem for finding quickly the effects of long-period and secular inequalities in the Earth's motion is simply quoted {Trans. Amer. Math. Soc, vol. vi.) ; this theorem was also used several times in Chap. XI v. ; but the brief derivation of the disturbing function for the motion of the ecliptic {Monthly Notices, vol. Ixviii.) is given in full. It was found to be impracticable to present within reasonable limits' much of the work actually performed. Numerous rough computations were made to find out whether coefficients or classes of coefficients were sensible ; when they were insensible, a simple note, often the result of clays or weeks of work, is made to that effect, but the organised plan of procedure always used in such cases is generally described ; this is the case with most of the results in Chap. XIV. Owing to the indications furnished by observation of an inequality or inequalities with a coefficient or coefficients of the order of 10" of arc and of very long period, one of the chief objects in view has been an investigation of such terms, and the " sieves" used in Chaps. XL, XII., XIV. were devised for this purpose. No large coefficients beyond those already known have been found. Moreover, the search has led more and more, to the conclusion that no such terms can possibly arise with the laws of motion and of gravitation on which this theoretical investigation is based. If these * Adams Prize Essay, Pitt Press, Cambridge, 1908. This will be referred to below by the letters A. P.M. Motion of the Moon. 3 inequalities have a real existence, it would seem that the cause must be sought in some action not purely gravitational. No part of the numerical work, except some of the multiplications of series which were necessary to find the derivatives with respect to n in Chap. X., Sect, (iii), has been turned over to computers. There are so many delicate points to consider, and so many terms and classes of terms have special peculiarities which permit the calculations to be much abbreviated, that to obtain the accuracy at which I have aimed by a general plan which could not permit these peculiarities to be used, would have involved an amount of computation out of all proportion to the final results. In fact, not more than one-third of the time occupied by these investigations has been spent on accurate numerical work. But all such work has been gone over at least twice, in many cases three times, and tested by comparisons and various methods, whenever possible. This part concludes the theoretical investigation of the motion of the Moon under the attraction of gravitation. Its natural sequence — the formation of tables to facilitate the accurate computation of the position of the Moon at any time or for the purposes of an ephemeris — has already been arranged for, and will be undertaken at an early date. But here also it seems advisable not to set the computers at work until an extended examination of methods which will best serve the purpose, and of the properties of the final results, has been made, so that the highest possible accuracy may be obtained within the limits set by practical necessities. The table of contents of Part V. follows. Chapter X. — Methods for finding the remaining Lunar Perturbations. Section (i). The equations of variations. Section (ii). Reduction of the equations to numerical form. Section (iii). Derivatives with respect to n. Section (iv). The final form of the equations of variations. Section (v). Numerical values of functions of the lunar coordinates. Chapter XL — The Direct Action of the, Planets. Section (i). The disturbing function. Section (ii). The computation of the coefficients P ; . Section (iii). The sieve. Section (iv). Numerical values of the elements. Section (v). The final results for the direct action. Chapter XII. — The Indirect Action of the Planets. Section (i). The disturbing function. Section (ii). The computation otSp', SV. Section (iii). Second method. Application to non-periodic changes. Section (iv). The motion of the ecliptic. Mr, Ernest TV', Brown, Theory of the Section (v). Numerical values of the Earth's perturbations. Section (vi). A sieve for the rejection of insensible coefficients. Section (vii). Computation of the lunar perturbations. Section (viii). Final results for the indirect action. Chapter XIII. — Action of the Figures of the Earth and Moon. Section (i). The disturbing function for the figure of the Earth. Section (ii). Numerical results. Section (iii). The action of the figure of the Moon. Chapter XIV. — -The Remaining Perturbations. Section (i). Corrections due to the masses of the Earth and Moon. Section (ii). The terms of the second and higher orders. Section (iii). Calculation of the terms. Section (iv). Perturbations with unknown constants. Chapter XV. — The Final Expressions for the Moon's Coordinates. Motion of the Moon. CHAPTER X. METHODS FOR FINDING THE REMAINING LUNAR PERTURBATIONS. Section (i). The Equations of Variations. 267. The Canonical Equations. — The problem solved in the preceding chapters may be stated as a solution in series of the equations dx, dt dll "dfd dy ,711 ay,- fin TT , , ., .-> ,->> ,-, -^ = - , 11 = Uy 2 + v., 2 + Y.. 2 ) - 1 dt (fa, ' 2 x " l ' 2 " 3 ; (1), where x, (i = 1, 2, 3) are the coordinates of the Moon referred to fixed axes, and F is the force function under the hypotheses stated in Chap. I., Sect. (i). Let w lt iv 1 — w 2 , iv 1 — iVg be the Moon's mean longitude, and the arguments of the principal elliptic term, and of the principal term in latitude, respectively, so that w. 2 , iv 3 are the "mean longitudes of the perigee and node'' resulting from the solution of (1). Then Jacobi's method shows that, if a quantity R, be added to F, and if it be expressed in terms of iv v u> 2 , w 3 and c v c 2 , c 3 , the other three arbitraries of the solution of (1), the latter three may be so chosen that the solution of the differential equations dc, dll ,hv : dR dt dw. dt dc, + h (2) will give variable values of the c { . w h which, when substituted in the expressions for the Moon's coordinates and velocities instead of their former values, will give the Moon's position and motion under the force function F+li. Here b 1} 6 2 , b s are the coefficients of t in the angles iv V/ w 2 , iv s . Hence, when .R = o, b 1 = n, the mean motion, and b 2 , b s are the mean motions of the perigee and node. They are functions of c x , c 2 , c 3 and the constants present in the differential equations; and b^dc^ + b^dcy + b^lc^ is a perfect differential. 268. Transformation to the Variables n, c 2 , c s , u\— -Change to the system n. c.-,, c v retaining the w ; unchanged, so that c 1 is now a function of n, c 2 , c 3 . Then Adn dc, dc, - l dc„ ) + b,dc„ + i)„dc, de 3 -J - A " ° is a perfect differential, and therefore, since b Y = n, d h - dc„ dA\ dn dc x dc. dn db 9 _ db B dc, dc wssmifBiam 6 Mr. Ernest W. Brown, Theory of the The equations (2) then become dn dR db a dR db,, dR dt a 2 fj\ du\ dn dw 2 dn di, dc 2 _dR W~dv„' dc $ _dR dt dm.. dw 1 _ 1 dR , dw„ It where dc„ ■' \ dt dn h \db 2 Viin ■■ div„ clR + b„ ■dio 1 lit' ,^8 dn (3), -a?(i , and all the functions are supposed to be expressed in terms of n, c 2 , c 8 , w t . It is to be remembered throughout that, when using the variables w { instead of the constant parts of those angles, the derivatives of w t with respect to n, c 2 , c 8 (or any functions of them) are zero. 269. Solution of the Equations.— It is supposed that R contains a small factor whose square may be neglected, and consequently that we may substitute the undis- turbed values of n, c 2 , c 3 , w { in R, that is, the values furnished by the solution of the main problem. In all the cases to be considered under this method, R can then be expressed as the sum of non-periodic and periodic terms, and each of these terms may be separately treated. First, for periodic terms, put R = A' cos (qt + q') = A' cos (i 1 w 1 + i 2 w 2 + i s w 3 + q"t + q") , where q", q'" are constants independent of n, c 2 , c 8 , w it and A' is a function of n, c 2 , c 8 only. Substituting in the first three of equations (3) and integrating, we obtain A' dq a 2 /Jq dn cos (qt + q') , 8a, ■■ i,A' i„A' •kt + q'), Sc :i = ^± COS (qt + q') , (4), where Sn, h 2 , Sc 9 are the additions to n, c 2 , c 3 due to R. The arbitrary constants due to integration are given zero values. If Sb v §b 2 , Sb 8 are the corresponding changes in \ = n, b 2 , b n due to these changes in n, c 2 , c 8 , we have <f6 x = fe, and from (2a) and (4) %h - dl> ^r,4- db ^ l - -u db 2S„ I 1 dq db 2 , dq\A' . t ,, B h = ^Sn + §S8 02 + p S( ^(^pp + ^ co dn de, 2 dr 3 \ a 2 f3 dn dn dcj q u J ' Denoting by Sw t the additional part of w { , substituting the value of R and these results in the second three of equations (3), and integrating, we obtain s 1 / 1 dA' A' dq\ . . , , x s W2 = { ('_L ^^_2£V +3 _ j_ d( i db M' I sin (af+r/) 2 S W/? dn dn do J q + \dc 2 a?ji dn dnJJ* } " m W ' + q) ' 8w„ 1 dA' db B dA'\i f dq i dq db,\A'\ . , , ,. (5). Motion of the Moon. 7 The equations (4), (5) constitute the theoretical solution of the problem. If X be one of the Moon's coordinates, then the additional terms due to R are given by 5,. d\« , dk« , d\„ , rfA.„ dk „ <lk ^ bk = ~--bn + — M:, + -—bc„ + , bw, + —— bw 9 + bw. . dn dc 2 ~ dc B aw 1 dw 2 " dw & 2 jo. The Constant Term of R. — Denoting it by R , we have, instead of the equations (4), Sn, Sc 2 , Sc s constant. These constants are at our disposal. Put Sn = S Q n, Sc» = o, Sc s = o. Then, instead of equations (5) we have dR n Sw 1 = a 2 /3dn ° bw 9 - f dR a db.)~ dc„ dn 6 V dc, dn V where the additive constants are made zero. Now, since the mean longitude is a quantity observed directly, we so choose § n that ■w\ is still represented by nt + e, and therefore Siv 1 = o. Whence the changes in the angles iv 2 , w 3 are obtained by adding to their motions the quantities 8/;.,= dR^ dc,. which include the change S n. Since Sw 1 = o, we have »0« = bb s = '//.' dR dc„ (6). (7). This change in n must be substituted only in the coefficients of the periodic terms representing the Moon's coordinates, amongst them, the principal elliptic term in longitude (2c with sufficient accuracy) and the principal term, in latitude (27). These, again, are quantities observed directly, and therefore, as we wish to retain the same expressions to denote these coefficients, it is necessary to add to e, 7 in all other terms the amounts 8 e = de „ respectively. It is true that these produce further changes in b 2 , b s , but they are quite insensible. Indeed, the changes (8) produce alterations less than c/ /- oi in any coefficient. The changes, as found from the methods of Sect, (ii) below, give V= - [3745 2 ] V . 8 y= - [3-88t2]S m . If R contains a non-periodic term of the form R p t p , where R p is independent of t, the corresponding changes in Sn, Sc 2 , Sc s are zero, and biv =— dE " f " +l 1 o?(i dn p + 1 Btv„ dR„ jp +1 ,db ?8w dc 2 p + 1 dn 1 ' 8w 3 = dR„ t" +1 , db, s dc s p + 1 dn Si pi 1 1 Section (ii). Redaction of the Equations to Numerical For m. 271. Computation of the c { . — The coordinates x,- of the Moon have been expressed in terms of e, k instead of c 2 , c 3 ; we must find the relations between the two sets of constants. 8 Mr, Eksest W. Brown, Theory of the Owing to the canonical forms of (i), (2), it is well known that 2,( yM + u Y lc { ) = dS, a perfect differential. Hence if every quantity be expressed in terms of c i} w if ^=S,yA f^ + ^% (*,; = !, 2, 3). dw< dwi dCi act But since T = dxjdt, these equations show that dS/dw { consists only of cosines and dSldct-Wt of smes of the angles present in the solution of the mam problem and therefore that 5-2^ + periodic terms. Denoting by [<?] the constant term of the expansion of Q as a sum of periodic and non-periodic terms, and substituting the value of S in the first of the previous equations, we obtain ' dXj dXjl ...... (9). 1 { ' dt died* the equations for finding c ( > The co-ordinates x x , x 2 , x 3 are here those referred to fixed axes. * e therefore put Xl + x 2 i - « exp. i(n'* + , x a - x 2 t = 8 exp. - .(»'* + , x 3 = a , to reduce to our earlier notation. _ Next, lt , s have been expanded in positive and negative powers oi I , I , i , { , Wh6re ii-erp.iK-»'*-0, «» = « P.^,--,). P = exp. .(«,-«-,), «--exp.^ + «), and numerical values have not been substituted in the exponents. In line therefore with the former definition of D, I put D,=i}~, A-Ki, D^i?±, V**-^' so that d A = t (z> 1 + A + A) , 4- = - lD - ' ^7 " " ^ ' D = D 1 + cD c + g-Dg + mfl m . Making these substitutions in the expressions for c it and remembering that, as we only need the constant term,/(«, s)+f(s ; u) can be written 2f(u, s), we obtain Cl = - In - ra' MD + i + m) h' • (A + A + A - J )«' + ^ ' (A + D < + AM) > } _3_- = r<z> + 1 + my ■ z>, «' + z> 2 ■ M » -~ 8 v = [(^ + * + m)rt ' ' ^ + Dz ' DA J where tt' = 'M^ _1 ) s ' = " s v , , ■ • As a matter of fact, Cl will not be needed. The values of c,, c 3 have been given m Chap. VII. , § H5 (there called ft, ft). They are ,^6^e% 2 -"oonoeV 2 , ) %= - '11844 440.2- -023246* - ^eaese-k 2 - 'oonoeV 2 > e„ _a_= - 2 002 .2 wa .05 9 k*- r 9 6 3 76k 4 - -285 4 6kV - -00568!^ ) The forms of these expressions are important : c 2 is divisible by e 2 and c 8 by k 2 , and if we neglect powers of m, c 2 = -\na\\ c.= - 2naV. The numerical coefficients are mmmmm Motion of the Moor to be considered as functions of n. It must be noticed, however, that in finding the derivative of c 2 with respect to n, the terms in m 2 , m 3 , . . . diminish the value obtained from the principal term by nearly one-half ; the derivative of c 3 is not much altered by these higher terms.* It is in general true that quantities depending on e, b 2 in any way are slowly convergent along powers of m, while those depending on k, b s are rapidly convergent. 272. Derivatives with respect to c 2 , c s . — These may be obtained from the deriva- tives with respect to e, k by solving the equations dQ dQ dc 2 dQ dc 3 de dc, 2 de dc z de . dQ dQ ,dc 2 dQ dc s dk dc 2 dk dc 3 dk where Q is any function under consideration. Inserting the numerical values of e, k obtained from §§ 192, 193 of Chap. IX., I find. adQ , r i dQ r - -,. dQ k =+[2S423> *^ [r773l]k "rfk' ;M 2^= +[2-o929]k^-[77386>^ dc, L J yj dk L '•* J de In § 145, Chap. VII., will be found the materials to obtain b 2 , h 3 (there denoted by ttj, Oj). From them I obtain, with the help of equations (12), de„ + [2-3i7S], « 2 -i^= + [2-3960] = a^ 3; fl2 p = _ [ 3. s698 ] . . (I3) . 273. Derivatives ofb 2 , b 3 with respect to n. — These might have been obtained by finding c x from the first of equations (9a) and using the first and second of equations (2a). But it was found to be much more simple and sufficiently accurate to use the existing literal developments in combination with the numerical developments. The method for doing this I have given in a former paper. t It is as follows : — Let /(m) = a + ttjlll + a 2 m 2 + . . • and denote byj^(m) the sum of the first i terms of this series, and by an accent the derivative with respect to m. Then m/'(na) = m//(m) + i i , * + ! >■ I * +1 ,1+2 i+o If i be not too small, and the series not converging too slowly, the error committed by putting m/'(m) = m//(m) + i[f(m) -/ 4 (m)] will be small compared with the true value of m/ v (m). When the series appears to be diminishing with fair regularity, the use of i + proper fraction instead of i in the last formula will probably give greater accuracy. As a matter of fact, I have only used this to find the greatest possible error which could have been committed, so as to avoid any sensible error in the result. The derivatives of & 2 , b % are found from the literal \ and numerical values of Hill * A.P.E., p. 8. f Monthly Notices, vol. Ivii. p. 346. \ Ann. of Math., vol. ix. p. 40. Royal Astron. Soc, Vol. LIX. 2 IO Mr, Ernest V. Brown, Theory of (he for the part of b, depending on m only, and of Adams * for the corresponding part of h, ; and for the other portions from the literal values of Delaunay + after a test by a special method which I have given earlier,} combined with my numerical values. These latter portions are expressed in terms of e, 7. Since the derivatives of e/e, k/V are insensible to the degree of accuracy required, we can make the change, after differentiating Delaunay's series, by the formulae of § 192, Chap. IX. The required derivatives with respect to n are obtained from equations (11) combined with d J* = ('' ( ±\ <!Q('!h\ 'I®(' Ic s) . . ■ (14). du \dn ) ~ dcXdn J ' dcS,dn ) where the brackets denote that the enclosed functions are expressed in terms of n, e, k. The derivatives of c,, c 3 are obtained from Newcomb's transformation § of Delaunay's literal values for G-L = c 2 , lf-G = o s , combined with the numerical values of c 2 , c, given above. They are only needed to two significant figures, jl The other derivatives with respect to n are found in the next section. These apparently complicated processes are constructed to avoid the slow convergence which occurs with certain of the literal series arranged in powers of m, and I believe they have achieved the object in view. 274. The subsidiary results are @^S-4637]«S G&h^-^K, ©)=-[-7o 9 ], ©H[3'5736], \dn and the final results dn l < j ' d n [2-1720], _^= +L3 5733J Section (iii). Derivatives with respect to n. 275. The equations for finding n-derivatives with the system n, c 2 , c s , w t .— The problem here is to find these derivatives from a theory in which the numerical value of n has been substituted^ I shall show that these derivatives may all be made to depend on derivatives with respect to the other five elements, and therefore with respect to e, k, iv t . Write v ('V #'_ i, f <ML\^\f f\ :,..,. (16). ^Adc, dw, dw, dcJ l ' ' '" Then, on changing equations (1) from the canonical set x { , y ( - to the canonical set c h vj { by a contact transformation, we have the relations * Monthly Notices, vol. xxxviii. p. 48. § Amer. Eph. Papers, vol. v., pt. 4 (1894), pp. 201, 202. t G.B., vol. lxxiv. pp. 19 et seq, || A.P.E., p. 8. % Monthly Notices, vol. lvii. p. 335. «TI E. W, Brown, Trans, Amer. Math, Soc, vol. iv. pp. 234-248. Motion of the Moon. Denote by brackets derivatives with respect to the system c { , for changing to n, c 2 , c 3 , w { are IV s df _ (df Y*i d f__( d fi\ d £i ( d f dn \defhi ' dc 2 Kdc-Jdr^ \dc 2 df fdf \cfej dc s \dcfdc s \dc if 1 1 The formulas . . (18), the other derivatives remaining unchanged. Write (see § 268) J^L= ( d Lj l !ft f£ _^£i l¥\^ ,lc j^(dK + d h fK + d b d f\^ (i f± , ( I9 ); dWf 2 \dw 1 dc 2 dw 2 dc 3 die, J ' dn \dw 1 dn <ho i dn dw % ) ' dn then {/,/'} transforms into (/,/'), in which n replaces c 1; and W x replaces w v It is convenient to change to our former axes by putting . , ,. / ,, i\ du ds x 1 + ix 2 = uex]i.(nt + e)t, x x - tx 2 = s exp. - (nt + e)i, x s = 2, M i = ^> s i = f ^' and the transformed equations which we shall need are (u 1 ,u) = o, (,//,s) = o, (u,z)-=o . . . • . (20). Now du x d du __ d du -y, /du_ dbA _ d du du <lcj dn~~dn ~di~di dn ■^- li \div i dn)~dt dn dW x dn ' If this be substituted in the first of equations (20), we obtain d du du dU jr 2 dt dn dn dt dc x du d du _ du d du du^ d_ du_ jlu_ d du^ dn dw 2 dt dc 2 dc 2 dt dw 2 div s dt dc a de 3 dt dn> 3 ' du db„ du db„ du - + - + ■ where JJ= t - , -- . dw x dn div 2 dn dw a Dividing by TJ % and integrating, we have the formula for du/dn : du -=U wh ere -/3« 2 = dn ' dn J\ U* J Q = l(xst four terms of (21), C= arbitrary constant. ") As du/dn can contain no terms proportional to the time, the constant term of the function to be integrated must vanish, and therefore - I -f- de. (22), dn [3a 2 which determines /3. The arbitrary constant C is determined from the second of equations (20), and the values of db 2 /dn, db 3 /dn have been found in § 273. The value of dz/dn is similarly determined from the third of equations (20) when du/dn is known. The process of finding the n-derivatives is therefore reduced to an integration. 276. Computation of the n-derivatives of u 2 , us, zu, and the products of these into (a/rf, — The process given in the preceding paragraph is neither simple in theory nor easy for computation. But, in the absence of any other method, it had to be >..,,..-,.... ......... w.i^.vMW.'K'.wr.V.v.v/. i 2 Air. Ernest W, Brown, Theory of the adopted. Various plans for abbreviating the work arose during the computations ; I shall not go into these details, bur shall only give the main outlines and formulae which were used. With the use of the notation defined in § 271, the definitions of U, Q will be slightly altered. It is convenient also to replace c 2 , c s by - a 2 me 2 ' 2 , - ahiki, so that e 2 /e and k 2 /k are positive. We put, then, '11 - = w£ _1 , f 2 = -« a W> c 8= -«a 2 k 2 2 > U-- = U r , + U, = (A + D,. + D, + i)u'- ^D,u' - C pD g u , 1 v l ' ** dn dn a= ,D,.u d T , , ,D,.Du du , Ti , d ,, , n n / e 9 de 2 " e 2 ae 2 " ak 2 ~ du dk 2 2 <£= ■^- I - m )' »^-^=^ + C >- ' • ( 2 3)- The computation of U, <ft is made in the following way. Let U consist of the terms of characteristic unity in U, so that U = {I) t + i)u '. Then 1 1 U-, U-, u 2 u* u* 6 r„ After finding i/U i by special values or otherwise from the results of § 44, Chap. II., and Ui by multiplication of series, and then U* for the few terms needed, multiplication of series and additions quickly give 1/U 2 ; the process is simple, since i/U l converges rapidly, and only three or four terms are needed. Next, Q is obtained by multiplication of series quite like those used in the earlier chapters, some of which could be made use of by means of the identity fDf —f'Df= D(ff') — 2/' Df. To obtain the derivatives with respect to e s , k 2 from those for e, k, it was nearly always sufficient to multiply by e/e 2 , k/k 2 , respectively, as the formulae (12) of § 272 show. Owing to the form of Q, it is necessary to use vl as far as terms with characteristics of one order higher in e, k than those needed in dujdn, but the results of previous chapters were far more than sufficient. Finally, Q/'U' 2 w T as obtained by multiplication of two series and thence /3, cp. The value of /3 as found here and tested by two other methods * is '32962. The second of equations (20) may be written du „ ds TT r, , dn dn du ds ds du du ds ds du _dc, div. 2 dc 9 div„ dc„ dtp, dc s div s . Denoting by a bar the change — 1 for i, so that S = — U, and using the new definition of U, we find 1 irffi n , a , j\ 1 1 Dm ds ,D,i du , ,, , ds ,. , du . , ^ (a c + * + ^^=i-^. _-i-.. _ + />,«. Si - 2 -^.- 5? . . (24 ). The multiplications of series for -^ were all at hand, and the computation of UU was simple ; to determine C, only the constant terms on each side were needed. * From the transformation of Delaunay's L = e 1 (§ 273), and from a theorem connecting it with the constant term of the parallax (E. W, Beown, Trans. Amer. Math. Soc, vol. iv. p. 247). ■" ■■-.<! • Motion of the Moon. To find the derivative of us = u's', the simplest form appeared to be wi-(Ms) = /^ - + u% = i(Y u + 7Tl)(<t> + 4> + 2 V) + 1 (V U - /U)(<j, - £) . cm <m «m " (25). since '"-'HM-s'M-^M" and the product ws was at hand, the constant disappearing, so that all the terms were small. For the purposes of verification (s'f/+. /£/)(<£ + <£ + 2 0')= j",+ (j! ^ (26) was computed : this gives the more important part of d(us)/dn. The derivative of u'' 2 is given by n£u'°- = 2u UU + U) = «- + G)(/J 1 + D c + IX, + 1 -- ^A, - '^- 3 A \P dn \ dn dn (27 for which the product u' 2 was available. A multiplication of series gives the required function. The formula for dzjdn is obtained from the third of equations (20). It is transformed into >i J f = Z(4, (- u ) - f* , z= Oh + A + A - ^ A - -V^ )« , dn db s , dn l D,x du_- [ l) K ii *L + /;-.. du _±D t ,iC dz_ " <'•? <'«•> de.-, //k 2 A k dk (28), the multiplications of series for x being at hand. The derivative of u'z is therefore given by n*Hu'z) = - %< + (* + C)f A + IX + 11, + t - ^A - ^ aV^ dn ' U A \ 'te dn 7 ( 2 9)> which is computed as before. Only the principal terms were required. The derivative of z l is given by (28), if we replace 2 by z 2 in the formulae. The few terms for derivatives of third-degree products of u, s, z were computed from these results. The numerical values are given in Sect, (v), 277. To find the derivatives of the products of u 2 , us, uz, with (n'rf, we only further need the derivative of the last. For the only term which is sensible we can use Delatjnay's results, combined with the numerical results of § 266, Chap. IX. This is because the constant term of a/r contains no portion depending on e 2 , y 2 , and the portions depending on m converge very rapidly ; the other terms in the products are of orders <r, y 2 , ra\ e 2 m 2 , nvy 2 , or of higher orders. The results from all but those of orders e 2 , y 2 , are practically insensible. ^^^^^^^^^^^p^tpp^^^^^ip^^^^^^^ 14 Mr. Ernest W. Brown, Theory of the Section (iv). The Final Form of the Equations of Variations. 278. Numerical values required. — I gather together the numerical results obtained in the preceding sections, so far as they are needed in the equations of variations. They are as follows : all numbers whose common logarithms are set down being enclosed in square brackets : — e= +[27396], e= +[1-0396], y=+[i-652i], k= +[2-6511], ] /3= +[7-51801], w = M = +[2-87391], m=- -- ,= +[2-90768], \= +[1-99921] I ■°" J ' : 11 n-11 a- J L^= -[2-1720], a? ^2= +[2-3175], a 2 Li?= +[2-3960], J Lj= +[3-5733], a2|*8 =+ [l- 39 6o], a? (] ^== -[3-5698] ! „dA' , r . -, dA' r- -,,dA' n dA' r -.dA' r - „,-, dA' -na-- = T [ 2 5423^ -[r 773 .]k^ -^ = +[2-o 9a9 ]k_ - [t 73 86]e_ (30, (32). 279. The Equations of Variations. — The equations are now put into a form which is to admit of direct application to any term of the disturbing function. The coefficient of each cosine or sine which constitutes the variation of an element is to be expressed in seconds of arc. Put j-x I III to 9 . ■ / . 4 . // , f. I. Tib ft) n a / 1 > -■ Jtt = — - , n ^& J A 00s (1-.W-, + im'„ + t„w a + t + a ) = -n i & i A cos (qt + ci ) . 4 m ' A L 1 1 ■ 6 6 1 j- 1 4 m > s' = number of seconds of arc in the daily mean motion of the Sun = 3548"'i9 , s = ,, ,, ,, ,, argument qt + q ; s'2 i m a- s 1 m a- tos Ql m /=— ^ — — 206265= +[ I 2-2935 8 ]— , /=————-= + [7-61748]^, 4 m a 2 (3 4 m' a 2 /3 ,, fA n f'A . . n d , , „> . , dA . , , dA =• — , G 1 =■>— , , }l A = - — (/la 2 , j,i = e T , ^1 = k - - a 2 (fo Aj = - i x + , oi486« 2 - -003744*3 , A,, = + -0148611 + -006624^ + -ooSadoig, A,, = - -003744*1 + - oo826oj 2 - -001238% /*,== -[2-i72o],/ 1 + [2-o6o3],7 2 -[i-29n]i 3 , \ /*:>,= +[3'57.^3]ii-[i" 2 S 66 ]i2 + [ I ' 6l °9]i3 i - = AjOj cos (gtf + q) , then the equations of variations are 8n •a 8c, na na- Sti'j = (AjC+ZjCj) sin (gf + q") . 2 2 — + [1 '5 1 801 JigC, cos (qt + g') , Sw 2 = (K,G + /.taCj) sin (qt + g') , -= + [i-5i8oi]» 3 C7 1 cos (qt + q) , Sjo 8 = (A s C+/x 3 C' 1 ) sin (</i + </') If it be desired to find §e, §y, they can be obtained from <33>; (34); \3Si ; (36), (37;: (38). te= -[1 2823]-^ + [3 43b ma a + [3-74S2] - > «>T= -L746o]^-[r22 32 ]— , + [ 3 -88i2]- (39). S:..' Motion of the Moon. i 5 These formulae are not easily comparable with those of Hill,* EADAU,t or Newcomb,^ because I use derivatives with respect to n on the assumption that A' is expressed in terms of n, c,, c s , while they suppose that the coefficient is expressed in terms of n, e, 7. The chief gain here is the avoidance of the doubtful derivative dc 2 /dn, which in my method is certain to the degree of accuracy required ; and further, the method in Sect, (iii) of finding the n-derivatives is more simple with the system n, c 2 , c 3 than with the other system. The derivative on which nearly all coefficients depend is dc s d.n = —f3a 2 , and this is found accurately. The comparison, however, can be made for all terms except those in Sw t involving C t : this has been partly done.§ 280. Method of using the equations and abbreviations. — The numbers f f are the same for all the direct perturbations of a given planet ; for all indirect perturba- tions ; they have two values for terms dependent on the motion of the ecliptic; and three values for terms depending on the figure of the Earth. The numbers \, X 2) X 3 , «.», //.;, are the same for a particular Moon argument in a given class of perturbations ; the first and third terms of n 2 and the first and second terms of m 3 can nearly always be neglected. Only two numbers, C, G x , have to be computed for each coefficient required, and one of these is generally very small. The most important particular cases are : — (a) i x = L — u = o. Then Sn = Sc 2 = Sc 3 = o and X,-C= o in the Sw { . (?;) Terms of long period in which %=|=o. The portions depending on C x are very small. (e) Terms oj short period approximating to a month or less. The portions depending on C are usually small compared with those depending on C lt and it is rarely necessary to compute Sc 2 , Sc 3 , owing to the theorem in the next section ; Sn is very small. (d) Terms depending on iv 2 or w 3 , but not on n\. These are much the most troublesome to compute, because Sc 2 , Sw u Sw 2 , or Sc s , Sw 1} Siv s , may produce terms of the same order in the Moon's coordinates. But even here Sn is usually insensible, and the theorem given in the next paragraph is almost exactly satisfied, so that the computa- tions reduce to finding Su^, Syj, (and Sc. 2 as a test only), or Sio t , Sw s (and Sc 3 as a test). 281. Substitution of the elements in the coordinates. — We have, for the longitude, 8 V «, + -,-■ - 1 )bw, + - bw + -- 61," \iiWj J dw 2 ' div 3 dV v dV, dV„ dV.. <•„ + — dn + -— <V.', + ^6 an dc, " tic. (4°) om Ihe terms arising from the first in this expression, Siv x , are called primary ; those fr the others, that is, from the periodic terms, secondary. || All terms in latitude are secondary, and there are scarcely any sensible terms in paralb ax. Aafr. Pap?,-*, vol. iii,, ami Coll. Worls, vol. ii. p. Ami. 01,*. Pari,!, vol. xxi. Cow, tie Inst. Pub/., Xo. 72. Ji A.P.E., p. 10; Monlhly Notice*, vol. Ixviii. p. 167. || A.P.I-;., p. 37. ■■ - ■ *-*■ ■**■*»* * ■ * • - 9 1 6 Mr, Ernest W. Brown. Theory of the Since Sn is rarely sensible, it is sufficient to obtain dV/dn from Delaunay's * final results. The values of dVjde,, dV/dc s are rarely required, except for the purposes of a test, owina; to the folio wins; 282. Theorem.** — If Sw % he confined to the term fi 2 C x sin (qt + q r ) and a coordinate X to the term psin (i z 'w 2 + ^- r ), where \f/ is independent ofiv. 2 and ij has the same sign as i 2 , and i 2 =j=o, * 2 =f=o, then the variation of the coordinate X due to §n\, §c 2 is dk o dX /-v-/-f,,/ •/. ,/i — bf 9 + — -dw, = [k„G,fi n s,m\qt + q -i„ w 2 - </>■ } . dc 9 aiO:, " ' "' That this theorem is true when we neglect all but the lowest power of e present in the coefficients and when Sn is negligible is immediately seen. For, in such cases, / 2 = |i 2 |, Sc 2 depends on dE/dw 2 , and Sw 2 on dR/dc 2 . But it appears to hold even more accurately. An exactly similar theorem holds for Sc 3 , $iv s , and the terms depending on w 3 in a coordinate. In the case of the principal term in latitude due to the figure of the Earth it holds within one-tenth of one per cent, of the whole. Section (v). Numerical Values of Functions of the Lunar Coordinates. 283. These functions are chiefly products of the second order in u, s, z. The computation of them by multiplications of series is quite like those necessary in previous chapters, and indeed the great majority of them had been obtained in the solution of the main problem. The formation of the derivatives with respect to n, e, k has been developed in previous sections. It is understood that the n derivatives are formed with j, G . char. = — — - (Ca 2 . char.) , &' an' where C is any one of these coefficients (given a numerical value here) expressed as a function of n, c 2 , c s , except in the case of the figure of the Earth terms which are formed withj^iC. char. =d(Cn z . ch&r.)/dn, and the ease of terms due to the motion of the ecliptic which are formed with j x s?C . char. = d(na?C. chav.) /dn. The few coefficients where an exact computation of these latter forms was necessary are given ; in general, these derivatives were not needed, or their values could be set down from the known principal term in the literal expansion of the coefficient. The values of (a/rf, (a/rf needed for the figure of the Earth terms, are obtained directly from the numerical values of ajr in § 266, Chap. IX. These are simple to compute for the terms needed, owing to the rapid convergence of the parallax terms. The formation of the derivatives has been explained in § 277. 284. The values of V, U needed to find SV, §U are obtained directly from the results of Chap. IX., after division of the coefficients there given by 206265. The fact that Sn is always small, combined with the theorem of § 282, and the expressions in §279 of Se, Sj in terms of §c. 2 , §c s , make the computation of the derivatives of V * Mem. de I'Acad, d. Sc, vol. xxix. chap. 9. t A.P.E., p. 15. Motion of the Moon. 17 with respect to e, y unnecessary ; they can, however, be found from the results of Sect, (v), Chap. IX.. if needed. The value of dV/dn so far as it is wanted is oiveii by n d :L^ +[2-0465] sin I- [2-4669] siu (2D - Z) - [2-3965] sin 2D dn (40- 285. The coefficients M { .— These are defined by the following equations, in which d is one of the angles and M t the corresponding coefficient in the expansions of the functions set down. Summations for all such terms constitute the complete values of the functions. According to the previous notation, V, V are the ecliptic true longitudes of the Moon and Sun, r 2 = us + z 2 = p' 2 4- z 2 , T = the mean longitude of the Sun, etc. Then r' 2 a 2 „ a" 2 p 2 eos ,„ T7 , N if, cos fl a 2 zp cos.,, , »\ M i sin . A/o cos a's p(r-' - 52 s ) cos p _ yl) M, cos ^ 1 ct* p 3 cos ( F _ r) = f 8 < ;os 6 ___ -5 ■ sin ^ r ; M.sin ' , r '3 a s sin - 51 ; i¥ 9 sm sin a'* 2(V 2 - { ,.'3 n 3 COS ™(V-h") = 1 It V ' M w sin e - J-/ 10 cos ' a^zp 2 cos F _ r iji a sin tf .'a ..a tin \ ; - i¥,„ COS -'3 .,3 ' sill For computation purposes these transform into a - us — 2Z- r' 2 - u l\ e T /' „.'■! „2 2\ ->- 3/ ,'2 a -2 cP u(us ~4z 2 ) e (T-vy = i( a/ 6 + 31 7 ) e ±^ , A . a '! "V 1 '- v "> = i(i¥ 8 ± Af 9 )e±«' 3 '' ° a a,' 2 IZ ■ (US - %Z 2 ) ,ir ,..>;. ,r . fl . a' 3 12 -it 2 ,., a - 10 ' r' 3 a 3 No values for M m M 12 are given below, as there were no terms large enough to make their accurate computation necessary. The computations of these coefficients from the results given are quite simple. Besides the operations already mentioned, there are multiplications by « /2 /V 2 , (a /2 /V 2 ) cos 2 V, etc., but the simplicity and brevity of these processes make further detail unnecessary. 286. The coefficients marked with a dagger (t) in the following tables include characteristics of two orders higher than the principal characteristic, and are therefore fully accurate to four significant figures. AH other coefficients were sufficiently accurate with the part depending only on the principal characteristic. Royal Astron, Soc, Vol. LIX. i8 Mi Ebxest W. Brown, Theory of the 287. Coefficient of C +u x characteristic = coef. of ^"" a ' : x char., in a 2 Char. ' = t=i j= - 1 1 + 99262")" - '00701! - '00701! e' m ~r O0787f 4- •00410+ - '02992! e' 2 2111 + 0101 t + '0005 t "'0878 ! e c _ 49°73t - '00262! -•08581! ee' c + m + 1455 t + '0016 t - '2090 ! ee' c - m - 1829 t -•0088 t + ■0338 ! ee' 2 + 2m + 0984 - '4034 ee' 2 c - 2m - 1817 - '0282 4- '2009 ee' 3 c + 3m + 01 -•85 ee' 3 e - 3m + 06 + •28 + '04 e 2 20 - 0618 - '0007 + 0758 eV 20 + m + 0371 4-40008 + '1794 e%' 20 - m - 0460 -•003 s 4- -0027 k 2 2g . + 2 9897 4- "0090 — '2062 kV 2g + m - 1092 -•0057 - 'S989 kV 2g - m + 0985 + '0409 2Z 2 a 2 + '2296 1 - 00803 + "00028 + '00028 e in - 00005 - '0003 3 + '00083 e + 00386 + '00014 + '00029 ee' c + m - 0014 - '0002 + '0007 ee' c - m + •0012 4- '0004 'OOOO k 2 2g + I 9997 + '0060 -■1479 k%' 2g + m _ •0646 - '0038 -•4377 kV 2g-m 4 •0762 + ■0273 d (us — 2Z 2 \ + '1608 e dA a 2 ; 1 + '00922 -•00057 -•00057 e' m + •00183 4- '00043 - '00177 e' 2 2111 + ■0021 + 0013 - -0042 For other arguments, d (US - 2Z i \ us - 2Z 2 de\ a 2 / a 2 with sufficient approximation. x index of e in char., n d — -r-tllS - 22") a 2 dn> ' -1-32383! - 0240 ! - '022 + "5925 + -56.19 — '264 4- -6oi + 02538I 01 5 1 j 000 0008 0070 0348 001 116 + -1- + 02538T 1 140 ! 354 2220 5i 6 3 °53' 99i 75 6 •0986 + -0008 - •2228 '100 -■003 - •440 •131 4- '012 _ •■t77 '9533 -■0310 + •4638 •018 4- -018 + -"5°3 •016 -•145 ~v T~\- 2Z & A dn x ' •526 •00800 — '00064 - •00064 •0001 + '0007 _ •0021 •0034 - -0030 - •°°5S -r 9 888 - -0208 + '3396 - '012 4- "012 + i"n6 - '020 - '096 - -376 d, 'us - 2» 2X a 2 / 62 2 k lk\ a 2 with sufficient approximation for terms k ^. I (US - 2??\ US - 2Z- lk\ a 2 index of x i k in char./ for terms containing argument 2g. IP Motion of the Moon, 19 2 88. Value of —>u 2 £~ ■ J a 2 Coefficient of £" ! ' 2i x chai xicteristic. Char. 8 1 = 2 i= 1 1 = •/ = - 1 i = - 2 1 I O + '0043! + '9879t - -0140! + 'OOOlf 1 e m - '0034! - -i86ot - '06l2"f + -ooo6t 1 e - m + '00021" + -oi82f + '19611 + '0056! - 'OOOlf I e' 2 2m - -0026! - 'io39t - -1817! + •003 it 1 e' 2 - 2m + '001 3! + "°553t + -i535t — 'oo66t e e + - oooit + -0046! + "4979t - '2982f + '0024"f I e - c + 'ooiof + ■1072! -i"4933t + '0123! ee' e + m - -0039 - "2342 - '7475 + '0167 1 ee' - c - m + •0057 + '3 2 34 + -1403 - '0090 ee' c - m + '0003 + •0186 + -303° + -0499 — -0019 1 ee' — c + m — '0009 -•0581 - '3 106 + -0576 ee' 2 e + 2m — "ii -i'53 + '07 1 ee' 2 - e, - 2 m + •03 + 75 + -13 ee' 2 c - 201 + •06 + '34 + "57 ee 2 - c + 2111 - '20 - '35 + -18 e 2 2C + ■0033 + '2507 — '2069 + •0259 1 e 2 - 20 + "0112 -■0519 + '6234 - '0002 e 2 e' 2C + 111 - -19 - '45 + •13 e 2 e' - 20 - m + •06 -■is - -24 e 2 e' 20 - m + '02 + '25 — 'II - 'OI 1 e 2 e' - 2c + m - 'OI + ■07 + '37 1 k 2 2g + '0030 - -1861 + ■0033 k 2 - 2g + '0005 + 0896 + 1-9743 - '0142 1 kV 2g + rn - '4i + '02 1 k 2 e' - 2g — m + ■23 + -28 + -oi k 2 e' 2g — m -•l6 - '34 - -06 IB 1 k 2 e' — 2 g + m Value of ~ 2 ^X u0 "'C 2 ) + -24 - 'OI 1 1 i - '00005 - -00806 + '00073 ,/ m + 'OOOI + '0046 + '0022 e - m - '0002 - -0050 — 'OO06 e' 2 2111 + '002 + 'OO5 ^■Ifilfl e' 2 — 2m - -ooi - '004 Fo • other arguments multiply by power of k iu characteristic. Bi Hi B i S^^l ... «.,.. , -ivwyA « ■ vawva ' wwp~ _ - ■■ gp^ 20 Mr. Ernest W. Brown, Theory of the d Value of -, -Wm^ -2 ). Coefficients of C +u x characteristic. Char. 9 1 = 2 i= 1 4 = i = - 1 -t= -a 1 - "oooi4f - •oi 4 54t - r3IS48t + •05410-!- - '00040T e' m + -O002 f + •0104 t + *355 2 t + •2420 f - -0040 f e - m - -0014 t - •0686 t - -3784 t - •0.254 t + '0006 t e' 2 2m + -116 + •820 - '022 e" 2 - 2m - "002 - T;4 - -234 e - '0002 - •°i33 - '6255 + •7981 — '0122 e — c - -0039 - ■2956 + 1*8414 - •0450 + 'OQOI ee' + m + '0004 + •0164 + -5468 + 2 •0102 - '0842 ee' — c - m - -0304 _ •9260 - '6502 + •0366 - '0002 ee' c — in - "0OI4 - •0738 - -8300 + •3428 + ■0058 ee' - c + m + '0030 + ■0148 + I c 3722 - •2232 ee' 2 c + 2m + '190 + 4 '344 - -364 ee' 2 — c - 2m - -140 •292 - -616 + •004 ee' 2 - 2m - •272 - 1 '064 - 2 '092 + '030 ee' 2 - c + 2m + •008 + •784 + 1-650 - •748 e 2 2C — "0002 - ■0116 - -2846 + *545 2 - '1028 e 2 - 2C -•0458 + •1216 - 7044 + •0006 e%' 2C + m + •016 + -458 4-1 ■130 -•540 e 2 e' - 20- m -•268 + •348 + 740 e 2 e' 20 - m - -002 - •070 - -676 + •976 - "022 e 2 e' - 2c + m + - oo8 _ •270 - 1 '306 + •006 k 2 2g _ •0002 - '0076 + '333 2 - '01 12 iv" - 2g - '002 2 _ ■1502 -1-9150 + •0456 - -0002 kV 2g + m + '004 +• •726 - -070 kV - 2g- m - '014 - ■374 - -484 _ •020 + - 002 k 2 e' 2g-m - -008 - •504 + '022 kV - 2g + m + '004 + ■366 e Value of -5 ar + -68o + •194 + '002 1 + -00003 + •00249 - -01584 + •00612 e m - •0021 - -0090 + •0176 e' - m + "00003 + •0082 + '0025 - ■0028 e" 2 m - •005 - -009 + •040 e' 2 - 2m + -oo t + ■020 + -004 - "012 For other arguments, multiply by power of e in characteristic. Motion of the Moon. 21 289. Coefficient of ^ x charac- teristic in w^- 1 -Hi n d a 2 a 2 dnS" 4 • 21 Char. Coef. k g + •998 - 1 •1646 k -g — 1 •0003 " k 3 -g + 1 '99 + ! ■1681 ke 2 - '5°3 k g+2 + ■0030 - •0108 k -g-2 + •0072 - •0246 k 3 -g-2 - •035 ke 2 -g-2 + ■063 k g-2 - •0457 + •1213 k -g + 2 + •°35S - •0857 ke 2 - CJ-+ 2 ! + '027 ke g + e + '503 ke -g-e + •496 ke g-e - i ■487 ke -g + c + •480 ke g + c - 2 - •280 ke -g-C+2 + •036 ke g-e + 2 + •108 ke - g + c - 2 + •088 ke g-C-2 + •026 ke -g+e+2 + •015 ke' g + m - '1082 + ■1439 ke' g + m - 2 - •1430 + •6456 ke' g- m + ■1176 - ■1755 ke' g - 111 - 2 + •0416 - •1805 ke' - g - m - •0835 + •1745 ke' — g - m + 2 + •1066 - ■2825 ke' -g + m + •07 1 8 - •1384 ke' - g + 111 + 2 - '°433 + '1084 ke' -g+m- 2 + •0299 - •1077 ke 2 - g- 2C - 2 - •0766 ke" 2 g+2C-4 + •0198 290. Coef of ?x char, in Char. ±6 Coef. I + I •0046 e 2 O + 1 •286 k 2 •00 1 2 + •01794 e 2 2 + •481 k 2 2 •00 e e + I '262 e c+ 2 + ■0427 e e - 2 + ■2504 e c-4 + •0081 e 2 20 + I '244 e 2 20 - 2 + •2208 e 2 20 - 4 + ■048 k 2 2g ■0000 k 2 2g-2 - "°39 Coef of ^ in a 5 (r 2 - 3* 2 ) r' a 2 + •994 2 + •0132 c + •0823 C - 2 + ■0158 20-2 + '00072 2g + '00613 2g-2 - •00036 Coef of '( e x char, in u l /a\ a Char. 1 2 + -9915 e c - '0697 e - c + 2 - '268 e c + 2 + 1723 e 2 -20+2 + '004 k 2 - 2g+ 2 + 2'OI3 » 2 0k = Coef. of £~ g in « 2 • ulT 1 ■ ?.i fa'v' _ ^ = - 1 '0062 t )\ = - 2 -1566 f h - + '012 1 7t h = + '99 2 ° t 291. Coefficients of £" in 2»£ l -D(fz) ka 2 ( 1 + in) _{D+ 1 +n\)(u'C'- a) ka -( 1 + m) Coef. g + '0O3 -g + 2 •0052"!' g+2 - '001 -g-2 + -006 g-2 + •°5 2 9 -g+2 + '0017 With factor «a 2 k and argument = - g •170-r /„ + '00026 + I'OOOOT h h k ;i^^^^^^^^^^^^^^j gppggi ppippjpg|gg pj| p p^ ^|jg^SPIgp 22 Mr. Ernest W. Brown, Theory of the 292. Values of M^ j^M,,, j 2 M i} j ?J M t . To obtain the values, each coefficient is to be multiplied by its characteristic, M lf M 2 are unaltered, and M s changes sign when the angle changes sign. Coefficients followed by the mark f have characteristics of two orders higher included. Char. Angle. Mi M 2 M 3 iJh kX hx t 1 + 99276"!" - '0280! - 1-3240! + -io8ot e r + 1 '0005 1 - '0835! - -0107! -1-3478! + '3 2 4St + -0509! e' 2 2 r + 1 2 59T - '201! - -040! - 1-701 + -852 + -302 e' 8 •7 + 1 •65 - 2'2I + 2'0 o'o e 1 - •49080! - '2862! - -3108! + '5926 + 7523 + '8423 ee i + i' - •3542-r - '421 - '477 + -18 13 + i - iii7 + 1-3085 ee' i-v - •6736! - '799 - -889 + I-I544 + 2-5564 + 2 9I24 ee' 2 1 + 21: - •370 - 73 - -8-> + '065 + 2-151 + 2-509 ee' 2 1-2I 1 - •978 - I'lO - 1 '34 + 1*904 + 3'595 + 4' 6 43 e 2 21 - •0618 - -2069 - -2065 + -0986 + '5453 + '544i eV 2I + V ._ •0247 - -24 — -24 - -ooi + -586 + -582 eV 2l-v - •1078 - 73 - 73 + -230 + 2 -6 1 4 + 2-604 1 2D - '00701! + "9870! + -9868! + -0254! -i - 3 I 44t - 1-3136! e' 2D + 1' - •00291! -1*17351' -1-1739! + -0103! + 1-6697! + 1-6709! e' 2D - 1: - •03693! + 3-1571! + 3-1561! + -1394-t - 4'3 2 4it -4-3169! e- 2D + 2l' _ '0042! + '0826! + -0818! + -017 - "239 - " 2 39 e'- 2D - 2 1' _ •1265! + 7-1584! + 7'i534t + -500 - 9-928 - 9-892 e l-2l) - •08582! - 1-4897! + i'4944'f + -2220 + 1-8274 - 1-8518 ee' I-2D + 1' - •2948! -4-321 + 4 - 349 + 7'383 + 4797 - 4-941 ee' 1-zD-l' - •0520! + n88 - 1-178 + -275I - 'S 01 + '44° ee' 2 I- 2D + 2/' _ ■7197 - '911 + 9-21 + 1785 + 9-01 -9-67 ee' 2 I-2D-2I' - •0716 - '03 + -05 ~ '425 + '24 - '31 e I + 2J) - ■0026 + '4972 + '4972 - "0008 - -6248 - -6246 ee' I + 2D + I' - '00 10 - 73 2 - 732 - -0078 + 1-1719 + 1-1717 ee I + 2D-I' _ '0114 + 1795 + 1-795 + '0340 - 2-7040 - 2-7038 e 2 2I - 2D + ■0758! + -6487! - '5969t - '2228 - -8064 + '6010 eV 2I-2D + V + ■ '2552 + 1-73 - i'53 - -663 - r8i + "93 eV 2I-2D-- V -4- •0785 - -18 + '32 - '400 - '93 + -27 k 2 2F + 2 •9901 - '2001 - -1717 -2-9549 + -3785 + '2873 kV 2F + r + 2 •8805 ^ ' 2 5 - -19 -2-937 + '5°9 + -275 kV 2 ¥ - r " r 3 •0882 - '37 - '27 -2-939 + -645 + "349 k 2 2 F - 2D - ■2062! + 1-9756! - 1-9690! + -4637 - 1-9243 + 1-9019 kV 2F-2D + Z' - ■8051 + 6'22 -6-i8 + 1-967 -6-282 + 6-164 kV 2F-2D-Z' - •1653 -2-31 + 2-31 - -319 + 2-583 - 2-605 Motion of the Moon. Values of M u JiMi, j\M,; j 9 Mi- -continued. Char. Angle. h.M, :kM, hM-i JJf, J s m, I o + '00922 + '01223 - '02415 + '00146 e /' + •oi 105 + '02710 - ■00403 e" 2 2l' 4- ' OT S4 + ■0413 + •0032 i 2D - •00057 - •01583 -- '01583 + '0008 - '0081 e 2D 4-/' - '000 [4 + •0685 + ■0685 e 2D-I' - •00235 - •04494 - 'O4490 e" 1 2D + 2I' + "oo 10 + '0004 + ■0004 e 2D - 2/' / - •0067 - •0918 _ •0916 + '01 16 JsMs o Char. Angle. M 1 M., M s ^6 M 7 J/ 8 M s a l D + '1132 + '3S°i + ■3681 + '972 + I'OOO - '007 - '007 a t e' D 4- 1' + '380 + '435 - ■194 - '195 a l e ' D - /' + 2'S°5 + 2-639 - •029 - '029 a^e I - D + •00396 - •3!363 + '26736 — 1 '464 + I'002 + •0139 - -0088 o.f.e' 1-D + V -3 '53 + 2-35 + •056 - '°33 a^k 2 3D - 2F + •0951 + '977 + •983 ~ r 57 - i'69 + •980 + -980 a,kV 3D-2F + // + '62 + '62 - 3 '48 -3'48 D~ 2 Z - •1387 + •179 - •131 o-je 8 3/ - 2D + •02080 - •01282 4- •06138 ajC 3 4D-3Z - '00226 ■- '02460 ~ ■02394 c^ek 2 I-2Y 3'497 + •2952 - •6828 a,ek 2 4D-/-2F + ■01108 + •01113 + '01 169 Ojk 4 4 D - 4 F •00 - ■0169 - •0165 Jhar. Angle. ih iA k 2 !0j - w 3 — h" + 1 •004 - i'i65 ke' 2W 1 - w 3 - h" + /,' + •896 - 1 '021 ke' 2 w 1 - w a - h" - V + t •Il8 - r 34° k w a - h" - 1 '0040! + ri68tt ke' w 3 - h" + 1' - 1 •08 7 1 + i'343t ke' w s - h" - I' - '93 2 t + i'03ot k 2T -w s -h" - '°4S7 + '1213 ke' 2T — ic H — h" + I' - •1887 + '7669 ke' 2T - w s - //" - V - •0041 - "°59 2 k 2 D + m; 3 - /i" + •°3SS - '0857 ke 2D + t« 8 - A" + I' + •1421 - -3682 ke' 2D + ?» 3 - h" - r - •0078 + '0227 Char. Angle, ek w x + w - ir B - It',' ek w^ — w„ + w z — li," k 3 - 2>i\ + 3?o s - h" e 3 k w 1 4- 3« 2 - w B - h" - 2I - 1 -49 + '49 - '989 - '0154 ... — ■ ■•■••ff^^.'.vAv.^vrv^^r»^.^^^9-'^-r,-'.i.'^T' '■'■'• '^ 24 Mr. Ernest W. Brown, Theory of the CHAPTER XL THE DIRECT ACTION OF THE PLANETS. Section (i). The Disturbing Function. 293. Axes and Notation. — The value of the disturbing function and the equations of variations are the same whatever axes be chosen. For the expansion of the former I take for plane of xy the ecliptic of 1850-0, and for axis of x a line parallel to that joining the Earth and Sun, that is, a line parallel to the Earth's true radius vector on the assumption that the Sun moves in an elliptic orbit. With these directions, x, y, z, r will now represent the coordinates and distance of the Moon, £, n, £, A those of the planet, the Earth being the origin. For the elliptic coordinates of the Earth and planet with the Sun in the focus, I take r', r" (the distances) ; V, V" (the true longitudes) ; T', T" (the mean longitudes) ; xs', vs" (the longitudes of the perihelia) ; o, h" (the longitudes of the nodes) ; o, y" (the sines of half the inclinations ; e', e" (the eccentricities) ; 2a! , 2a!' (the major axes). All these longitudes are measured in the usual way, that is, from a fixed line in the plane of xy to the node, and then along the plane of the orbit. The perturbations of the planet's orbit, like those of the Earth, axe neglected in this chapter. In order that the motion of the Earth round the centre of mass of the Earth and Moon may be taken into account, the terms depending on a/a' in the disturbing function must be multiplied by the ratio of the difference of the masses of the Earth and Moon to -their sum (Chap. I., §4); we must therefore use a x a/a instead of a/a' (Chap. IX., § 193), or, with sufficient accuracy, a t instead of a. The masses of a planet and of the Sun are denoted by m", m', and the mean longitudes of the planets, measured like the other longitudes, as follows*: Mercury, Q ; Venus, V ; Earth, T ; Mars, M ; Jupiter, J ; Saturn, S ; the other planets and the asteroids will not be considered in this chapter. 294. The Disturbing Function R and its Transformation. — -From Chap. I., Sect, (i), we obtain, on changing from the Sun to a planet as the disturbing body, R ....... x _x^ + yr l + zt A 2^p + v 2 + p . . . (1), to" {{£ - xf + ( v - y) 2 + (£ - zf}* A :i * Mean longitudes are denoted by roman capitals. t A.P.E., sect, it, where the full details of the transformation will be found. Motion of the Moon. If we denote by d/dQ the operator, 3 3 3 3 l)Q of o-q o£ the expansion of R may be put into the form R I C A _ I 0" 2T 3CP~ 3! 3a 3 ' or , since 1/ A is a solution of Laplace's equation, so that 0" I id a into the form 3f 2 St; 2 / A — ).?-- + terms of hieher degree \c'£* oj-v \c£- 07]-/ ti^crj \ ct crj/ oQ Now, in polar coordinates, we have £ = _,-' + (!_ y"2) r " cos ( F" - V) + y" 2 r" cos ( V" + V - 2I1") , 7] = ( 1 - y" 2 )v" sin ( V" - V) - y" V sin ( F" + V - 2/1") , £ = 2y"(i - y" 2 )V" sin ( V" - h") Hence 3/3£ = - 3/8/, and if/ be any function of if + >? 2 and £, 5/' ,3/ ,3/ , 3/ ,3/-' „—} = - r X- ~ k-- + '1X-- = ~ r X > 3 V 07] ' 07] Ct 07] 00 : (3), JL ( 4 ), A (5), r 3 V"- ?■-— + < d?; 0?; 3\5/' ,3 2 / c f Xof c/'\ of ,c 2 f 3/ o^/ct] C7f CTj\ ct] C£/ eg c^ cr ?l_i!_. ^'-JL ^ iS^. 8 2 /_5 / / i 3/ >2 ~ S,.'2 ' TXXi ~~~ Xa> -XXX'i + „.' 2,.' Hence 3f 2 3r' 2 ' ct; 2 '; /2 3 V' 2 ' r or ' o£otj cr'\r' oV Further, since (cr/dp + <f/d>f)f is a function of £ 2 + >? 2 , £, only, crj\o£ 2 erf J r 3 l"\3r' 2 r' 2 V" 2 r' or' (6). (7), and since we may differentiate each of the equations (6) with respect to £, that is, to - /•', we can find all the derivatives of the third order with respect to £, n in terms of derivatives with respect to r', V ; and so on. Again, by expressing A in polar coordinates, I have constructed the formula * ° I _ _ "/' e< v '- ]i ") t 3 1 3 , _ » 2 \_^ I l U, A" " {T-^y^y 1 ~~i-'"~ \ 3F' + S/ 2 3/7 i l " y V' 2 J ^ (8), which may be combined with the previous equations, since the left member of (8) shows that the right-hand member is a function of if 2 + i? 2 , £, and that its imaginary part is zero. * A.P.E., p. 19. Jaoyal Astron. Hoc. Vol. LIX. ^^w^^^^^W^^^^PPPP^PPPPBPIll! 26 Mr. Ernest TV. Brown, Theory of the Finally ,3,333 , x cr oa cv el and therefore if i/A be expanded in terms of the elliptic elements of the Earth and the planet, the functions needed are all expressed as derivatives of i/A with respect to the elements present in the development. It is true that these derivatives are to be multiplied by 1/r' 2 and that the lunar coordinates have to be transformed so as to be referred to the true instead of the mean place of the Sun ; but the work needed to perform these two operations is very small, especially when compared with the labour of making developments of several different planetary functions, such as A3 a 5 ' A 5 ' /\ s ' etc. 295. The transformed Disturbing Function. — I omit the algebraical details necessary to follow out this method, so as to present the results in a form convenient for numerical application. The result is a form for the disturbing function expressed as a sum of products : the first factor in each product is a function of the Moon's coordinates, u, s, z, multiplied by a certain function of r' , V ; the second factor consists of derivatives of i/A with respect to a', T, h", y //2 . The notation for the first factors has already been given in § 285 of the last chapter. If an angle 9 is present in the first factor, then the corresponding term is M t cos, 6 + iMi sin 6, where such constant factors have been taken out that the M { may be numerical quantities. 296. For the second factors, let a term with argument <p in i/A be — = P COS 1 If i, i! be the multiples of T, 2h" present in <p, the operators 9/3 V, djdh" give rise to the factors i, i', and the cosine is changed to a minus sine. Also P is of degree - 1 in length, and may be expanded in the form f{a)/a', where a = a'la", or in the form f(a)ja", where a = a"/a', according as the orbit of the disturbing planet is outside or inside that of the Earth. If, then, we put /= ad/da for a derivative with respect to log a only in so far as a occurs explicitly in P after P has been expressed in one or other of these two forms, we have : (11). When we combine the two factors, we obtain the product of two cosines or two sines multiplied by a constant factor ; this product is expressed as the cosine of the sum or difference of the two angles 0, <f>. The notation for the planet coefficients is as follows : — For outer planets, r dP d = a-r" = I , da ' a a = — , P= 'k&-> For inner planets, d d = — a.— - 1 = da -I- 1, a" a _ a ' P-- =>o Motion of the Moon. Let J + , »/_ denote the operators 27 oy - \ y "/ and let *', * 7 denote the multiples of T, 2W present in <f>. For outer planets put P^iP-i^P, P^l-P^-lP + PP, P 3 = -i(l- i)P, P 4 ±P 5 = -(l±i)J±P . P a ±P 7 = (I-2 + i)P 1 , P B ±P g *=(l+ 3 i-6)P 1 + 4(i±i)(2±i)(I+t)P, P W ±P U = J±( - A + 2lP±2iP) , P l2 ± P ]3 = - / ± {Pj + ( ± 2i - 2)IP+{2t l + 2«')P} all the upper or all the lower signs in any equation being taken together. The product of the pairs of factors gives for any term of argument 6±(p in R, (12), (ISA ,, Ilk m ; ,1 R = — n -&- ■ aa 4m M.P, + M 2 P 2 + M S P S - ^7 M^ ± P f) ) + \a x (M 6 P 6 + M 7 P 7 + M S P S + M 9 P 9 ) - y a \,.,. {M, (P w ± P n ) + M 12 (P 12 ± P 13 )} 2(l-y')= cos(0±<fi) . (14), all the upper or all the lower signs being taken according as it is convenient to use Q + (p or 9 -cf>. For inner planets, we replace aa by a' in this formula, and /by —I—i in (12), (13). It is to be noticed that Pfi" for outer planets and Pft' for inner planets are numerical coefficients when the value of a has been substituted. 297. Method of computation of the coefficients in R. — The terms in R with suffixes i, 2, 3 give rise to nearly all the sensible perturbations of the Moon's orbit. In the great majority of cases M % P^M^P Z is small compared with either of the two terms, and this is due to the approximate numerical equality of M 2 and M z and of P t and P s . It is therefore better (and the computations were so made) to use these two terms in the form i(AT 2 + M :i )(P 2 + P s ) + i(M 2 ± M t )(P 2 - P s ) ; and then again one of these two expressions was generally small compared with the other. Another advantage of this form arose from the fact that it was sufficient to have M 2 , M B to four significant figures and JP 2 , P s to six, instead of both to six. And, moreover, the near equality of P 2 , P s can be foretold by the theory, when it exists. A similar circumstance holds with the pairs with suffixes 4, 5; 6, 7; 8, 9; 10, 11 ; 12, 13. There is not a large number of terms depending on the terms in R with suffixes 4. 5, fewer still with 6, 7, 8, 9, and none sensible with 10, 11, 12, 13. It did not seem necessary to carry the computations to the next term of R, depending on a x 2 ; this coefficient gives a factor 6 x icr 6 compared with the first terms of R, and it gives rise to no arguments which are not present in the first terms. 28 Mr. Ernest W. Brown, Theory of the Section (ii). The Computation of the Coefficients P { * 298. Leverriers Expansion of 1/ a. t— Leverrier's literal expansion in powers of the eccentricities and mutual inclinations, with coefficients depending, on a and arguments on T, xs', m", h", was used. Here the Earth is supposed to move in the plane of reference, and the notation is slightly different. I have, therefore, put in the development — For Leverrier's symbols, t, t', /', X, <■>, a, «■, V the symbols h", //", 'J', T",i ~", «", a, y for inner planets, and have then interchanged the accents of a, rs, e, T for outer planets. Leverrier's development contains functions of a through certain coefficients (3%, which are defined as follows. Put A", = 1 + a- - 2 a COS (T - T") , j . = W2" OC j8--'cosi(T-r') ) <C= ^:£#' \ ■ ■ ■ (is)- i_i A," 2 ,: -co ' ■' p\ da Instead of ft, he uses the letters A, B, C, D, ff, //,§ according as s = J, f , f , £, f, V; I adopted the same notation, as well as the following : H«> = t^/J"- 3 ' + 9D"- 1 '' + 9 Z>''+ l! + L> ii+3 <) , L"» = |(C ! '- a + a" 1 ) , to which the suffix p may be attached according to the previous definition. But I dropped the brackets round the indexes to the letters, since powers of these functions do not arise ; and the indexes themselves were also dropped whenever they were all the same, equal to i, in a given equation. 299. Formula for computing the coefficients.— Leverrier gives the numerical values in most cases up to i=io, but they are needed much further in many cases of the lunar problem. Hence, all of them were newly computed by the formulas which are fully set forth in Section III. of my Adams Prize Essay. These formulae are constructed for several purposes : first, for finding isolated coefficients for special values of i ; second, for making tables of coefficients for many consecutive values of i ; third, for the avoidance of those small coefficients which appear as the difference of two large numbers. The third point is a difficulty which arises chiefly in computing (P-i*)^, the two parts of which are large compared with their difference, especially for large values of i ; but the formulas for these completely surmounted the difficulty. * A.P.E., sent. iii. t Aim. Uhs. Paris (Mem.), vol. i., where the expansion is given so as to include terms of the seventh order with respect to e', e", y". Boqukt {lb., vol. xix) has computed the terms of the eighth order. I Denoted P in A.P.E., by inadvertence the same letter as for the general coefficient. § Leverrier does not need the last two. Motion of the Moon. 29 It arises in a less troublesome form in P 2 , P 3 , but there the number of places of decimals computed was always sufficient for the degree of accuracy required. 300. Numerical values of A' p , B],, . . . — These were computed to six significant figures in tables * as follows : — o Venus : A I Bl to i = 43 ; A), Ii), to p = 4, 1 = 30 ; C), to p = 2, i = 30 : D M L\, 3"„ to i = 30 ; { (/" + i) 9 - * 2 ML t0 i J = 3> *'= 3° ; {(/+ i) 2 - (i+ i) 2 }-#j, top = 2, from * = -30 to i= 30; {(/+ i) 2 -i' 2 }C"', to ji? = i, 1 = 30. Jupiter to i = 6 for A), to p = 3, B\ to _p = 3, 0), to p = 1, Z>j. Mars: A\, BI 10 i = 30 ; and to « = 6 for A\, A\, B[, Bl,, C;. Mercury to i=-- 8 for .4],, i?]„ to /; = 4, Cj to jo = 2, Z>j. All other coefficients required with these planets, and those for Saturn, were separately computed as the needs for them arose. Section (iii). The Sieve.* 301. The larger number of the terms in Ii which give rise to sensible coefficients in the coordinates have periods which are comparable with the month or the year. For such terms the obvious plan was to take the successive values of 6 (the argument arising from the lunar factors) according to the magnitudes of the coefficients which accompanied this factor. The arguments o, I, 2D - 1, 2 1), . . . were successively combined with all the possible arguments <f> until the terms became insensible and it was unnecessary to proceed further. In each case the argument <j> was divided into the series i(T — V), f(T- Twits', and so on, the magnitude of each series mainly depending on the power of e' , e", j" which accompanied it ; and again each of these series was computed with a sufficient number of values of i with each Moon argument. A little practice quickly enabled one to choose out the largest coefficient in each set, and a rough calculation was sufficient to show whether the term would be sensible. This rough calculation had to be made for both hv x , $w 2 . and sometimes Sio 3 , for, with terms of short period, the secondary inequalities (§ 282) are frequently sensible when the primary might be neglected. This method could not fail as long as the periods were short. These periods only arise in the equations of variations (§ 269) through the divisors s, s 2 . But if any period was long, then s, s 2 would be comparatively small, and a large coefficient might result. Such cases as occurred during the progress of the calculations were naturally dealt with as they arose ; the only matter which called for attention was the necessity for a larger number of significant figures in the coefficients M it P { . If, in the general method, the numbers were not sufficiently accurate, the special coefficient was separately computed again with more places of decimals. But the lunar terms contain multiples of four different arguments, w 1 — T, iv 2 , iv 3 , T-tir', that is, combinations of four different periods, aud the planet terms two periods, 1 '. T. Hence we may have combinations of five different periods, and there will be * A.P.E., sect, v., where the numerical values will be found, * A.P.E., sect. iv. ,v*.f7.K'*.'*rfC>S.::- f A.9<S'7s!>rwi'.'9WfXy- 30 Mr. Ernest W. Brown, Theory of the many long periods arising therefrom. It was therefore necessary to sift out those which would give sensible coefficients. 302. The Sieve. — The method was essentially the same as that for the short-period terms, but, as there were thousands of possible combinations, some plan had to be devised to find an upper limit to each coefficient so rapidly that every coefficient might be examined within a reasonable time. The limitations were as follows. Only primary terms were examined. It was shown* that the secondary would not be greater than the primary unless i x = o (§ 279), s>6o" or i x =i, s>iooo" ; the very few of the former were separately examined, and the latter had been treated in the short-period terms. Periods greater than 3500 years (s<i") or coefficients <o ;/ 'Oi were to be excluded ; neither could sensibly affect the motion of the Moon within historic times. But one or two longer periods with coefficients greater than o"'oi which appeared in the course of the work were retained. The possible long periods were then constructed by finding all up to the largest values of i, j in i(T - T") +jT, ±w x + i(T - T") +jT, ... It was soon seen that only a dozen or so in each set need be retained, and at the most three multiples of w x . Then a table was formed for the multiples of u\, w 3 , giving the periods, the lowest orders with re- ference to e, k which would accompany each multiple, and the multiple of T which would occur in the lunar argument with this lowest order. Thus, for a given multiple of w % , iv s , the various long-period combinations with the former sets could be seen at a glance. 303. Next, very simple formulas were constructed for the primary coefficient, depending only on the power of e, k present in the lunar factor, the multiple of T-T" and the coefficient present in the expansion of i/a s or of i/A B . These formulas arose from transformations of the disturbing function somewhat similar to those of § 294, but depending on derivatives with respect to T only. The values of the coefficients in the expansions of A" 8 , A" 5 were obtained from Newcomb's tablet of these coefficients in the case of Venus; partly from his incomplete table, + and partly by extrapolation and by approximate computations, in the case of Mars ; for the other planets, which presented little difficulty, a table for A 3 was roughly computed, and simple formulas depending also on the order of the eccentricities and inclination were constructed. The various coefficients were examined according as they arose from terms whose characteristics were of orders o, 1,2,... With each order was associated a maximum value of s which could give sensible coefficients ; after the first three or four orders the work went very rapidly, as these maximum values of s became small, and the great majority of the terms could be excluded without computation. About 100 long-period terms were retained out of several thousand examined, and their coefficients were accurately computed. In no case did these coefficients exceed the preliminary estimates found by means of the sieve. No new terms of any great * A.P.E., p. 38. t Wash, Astr. Papers, vol. v., pp. 248-257. 1 L.c, pp. 258-261. Motion of the Moon. 31 importance were found, and the corrections to Radau's values* of those previously computed were small from the observational standpoint, as far as the long-period primaries were concerned. Section (iv). Numerical Values of the Elements. 304. Most of the observed quantities required are known with more than sufficient accuracy. The most doubtful is the mass of Mercury, which may be in error by as much as 50 per cent. ; but the largest coefficient with the adopted value is less than c/'-oS, and the term has a period of 39 years. The mass of Venus may be in error by 1 or 2 per cent., giving a maximum possible error in the largest coefficient (period 273 years) of o //- 3 from this cause. The values of the elements used are shown in the following tables : — Arg. i Daily motions Epoch Longitudes at Epoch. of arguments. : 1850-0. Perigee. 75° 07' 19" 129° 27' 34" 100° 21' 40" 333° 17' 55" 11 54 27 90° 06' 40" ]S T cde. w 1 iv 2 w 3 ! Q V T M J S 47434-89I 400-923 - I90-772 I4732-420 5767-670 3548-I93 \ I886-5I8 299T29 120-455 Mercurv 46° 33' 12" 75° !9' 47" 48° 24' 01" 98° 55' 58" 112° 20' 51" Earth Saturn Moon e = ... „ e= ... Earth Eccentricity. Inclination. Sine halfinclin. 1 «" ■m' m" ' ' I0 955 •054906 •016772 •205604 •0068446 ■093261 •048254 •056061 7° 00' 07" 3° 23' 35""3 i° 51' 02" i° 18' 42" 2° 29' 39" i k = '044780 •y = '044887 '061066 ■0296063 '016149 - oi 1466 '022 loga! = 3"39 88 ¥•5878216 i'8593374 '1828960 ■7162374 '9794957 6000000 408000 309350 1047 '35 3501-6 Mercury Venus Mars Jupiter L.e. (§ 279), p. 113. ?•'•'"•-• --^r>"w^^7^.-^rj?;-^^ 32 Mr. Ernest W. Brown, Theory of the Section (v). The Final Results. 305. The detailed results arising from each term of the disturbing function are fully set forth in A.P.E., Section vi., and they will not be reprinted. Many of the resulting terms in the Moon's coordinates, especially those arising from the short-period terms in R, have the same arguments, and must be combined. The final results only will be given here. There are two methods of expressing the perturbations. The first is to add them to the true longitude, latitude, and parallax of the Moon ; the second is to leave them as additions to the elements w v iv z , w z , a, e, y which would be tabulated with these additions. This latter method is only of special advantage for tabular purposes when the variations of a, e, 7 may be neglected, and this happens only with terms of long period in which u\ is present. If %v 1 is absent from the primary, and the period of the term is not very long compared with the periods of the Moon's node or perigee, the variation of e produces an effect of the same order as the variation of iv 2 , and that of 7 as iv n ; in fact, the statement in § 282 has to be remembered. In these cases the variations of a (or n) are insensible, and it may be convenient to retain the variations of tv x (or of iv v w 2 ) as elemental inequalities, adding the parts due to the other elements to the coordinates. No periodic variations of e have been retained as elemental inequalities. For certain other cases in which w 3 is present, with §e, 8a insensible, it is best to retain Sw lt Sw. it $w 3 as elemental terms, and account for those arising from Sy by multiplying the final value of the latitude by the variable factor 1 + Sy/y, and the terms in longitude containing the argument 2F by 1 + 2^7/7. But in setting forth the results such terms are left as perturbations of 7. Hence the terms are placed in two classes, those added to the coordinates, and, in addition, those added to the elements. The original limit set was o //# oi, but all short-period terms and most of the long- period terms have been computed to o" - ooi, and they are so retained here. A star replaces the last figure in the cases where the computations were only made to o"-oi. 306. The tables are arranged according to the lunar arguments so that 6 remains the same until a new value is set down, and then according to the multiple j of T (or j" of T"), which again remains the same until a new value is set down, and finally, according to multiples of T-T". The coefficients are set down in units of o"'ooi, the angle a being so chosen (<36o°) that they are all positive. The value of the angle a is also not generally repeated when it is the same for a long series of terms. Motion of the Moon. H 307. S V = + 0" -oo 1 sin { 6 +JT + i{T - V) + a } , Venus. = B=2J) 6 = 2D 3 i a 3 i a J i a c IH^hI o I o'o 480 0-15 I -1 - s 84 7 ifl^B 2 200 - 14 2 - 4 78 7 IIH 3 92 - 13 2 - 3 4 Ufls 4 60 - 12 2 - 2 4 iH^H S 38 -" 3 - 1 3 MHI 6 25 -10 5 1 1 I^^^^H 7 l 7 - 9 6 -2-6 162 6 I1H 8 12 - 8 8 i5 'S 1 4 ll^K 9 § - 7 8 18 IS 1 10 PHH 10 6 -6 11 11 4 - s 11 6» =/ 12 1 - 4 10 0-8 i8o - o 2 i[^^^Bl 21 3 - 3 i go 36 - 7 4 ifl^E . l - 3 92-2 1 - 2 26 - 6 il^E - 2 4 - 1 15 - 5 6 fl^E - 1 8 1 15 - 4 9 liH 1 47 2 8 - 3 l6 B^^^^H 2 272'2 76 3 4 - 2 29 l^^H 3 2I 4 4 - 1 68 i:flH 4 12 5 4 1 O'O 9i ;:'^H1 5 7 6 3 2 64 ili^^Ba 6 6 7 3 3 i8co 127 it^^^Hi 7 4 8 3 4 7 l^HI 8 1 . 9 2 5 1 ll^^^Hl 2 - 18 209 50 10 1 22 2 {f^^^HI - 4 27 i 18 3 1 1 92 8 - l^BI - 3 2 1 20 273 3 2 272 13 IBBI - 2 2 -1 -15 78 1 3 6 IIBI - 1 3 - 14 1 4 8 l^B 1 6 -■ 13 2 s 92 4 I^BB 2 8 - 12 2 6 2 I^^HI 3 37 - 11 3 2 3 272 6 l^HI 4 207 8 - 10 4 -1 - 5 268 1 ^^Hl^ 5 3 - 9 4 - 4 Hi HB 6 4 - -8 4 - 3 i:I^P 7 1 - 7 5 - 2 13 I^BI 3 5 112 7 - 6 s - 1 88 8 H^Hl Royal Astron. Soc, \ OL. -Lil-A. 5 34 Mr, Ernest W. Brown, Theory of the SV= + o" -oo i G sin { 9 +/T + i(T - V) + «° } , Venus. i a C I 3 280 7 2 -18 209 3 2 27 1 3 6 4 207 1 6 198 16 2 - 4 333 1 - 3 153 6 - 2 1 e= = 2 D-Z O -*3 iSo'o 1 — 12 2 - II 2 - IO 3 - 9 6 - 8 8 - 7 13 - 6 22 - 5 39 - 4 87 - 3 716 - 2 O'O I S 2 — I 74 I 13 2 10 3 7 4 S 5 3 6 2 18 11 19 2 I - 1 310 2 I - 10 269 1 - 9 2 - 8 3 - 7 S (9 = 2D-Z i i a G — 1 - 6 269 8 - 5 2 5 - 4 % 33 - 3 10 - 2 S - 1 3 17 253 3 - 2 - 7 340 3 - 6 162-5 83 - S 165 4 15 151 25 e = zT> + l - 9 1 - 8 2 - 7 2 - 6 2 - 5 3 - 4 1 - 3 4 - 2 4 _ ! 3 I B=2l 1 - 3 180 1 — 2 2 - 1 S 1 s 2 2 3 180 9 d=2l-2D 1 i8co 4 2 11 3 76 4 3 :2Z-2D i l a c 1 4 92 r 2 6 17-5 = sZ-4D 65 3 8 2 6 18 6= -4D 2 3 6» = 3 Z-2D 7 3 180 3 2 6 18 6» = 2F-2D 2 3 180 2 20 2 1 3 273 5 1 ~ 3 87 = Z-D 27 1 3 93 = ^ + D 40 1 - 3 87 2 s 355 3 4 9 3 75 16 2 5 = «>3+2F 75 Motion of the Moon. 35 ihb 308. sv= = +o"-oo2 sin (2D + T — 3Q+ 105 ), Mercury. fif^^^^^H^n^^^l 3°9- sv= + o"-ooi C sin {0 +/M + i(M - T) + a }, Mars. 6-- = 6>= 2 D 6> = = 2D-Z j" i a c j" i a c j" i a ' ^^bEH I 19 -1-5 "49 3 -5 180 '■m III 2 8 - 2 - 6 297 2 -4 17 ^BlU 3 3 -3 5 HNII 4 2 = Z - 2 : : ■ HM 5 1 - 1 180 2 -1 -7 149 1 ll^^^^HI^H 1 o 2I2'7 5 102 -6 ^^HH i 60 4 180 3 -5 ^■H 2 33 10 1 1 212 7 -4 3 2 9 3 II^^^HIBIH 3 4 2 32 1 -3 ^^^^BS^H 4 2 5 212 8 -2 -6 297 ^HIH 5 1 - 1 - 2 147 1 2 i 243 2 -1 327 7 = 2Z-2D 2 12 2 2 243 1 4 180 ^^^■■S^l 3 63 5 6 63 6 1 5 211 ^HH 4 2 - 2 - 2 297 1 2 6 243 ^^^■H^l 5 1 3 3 276 2 6 = 2D- I 6> = 3Z-2D 4 96 1 0-6 180 1 ' 2 6 243 ^^HfEliH 310. ,57= + 0" -oo i C sin { +/' J + i(J - T) + a °}, Jupiter • $■■ = (9= 2 D 6 = 1 f i a c /' i a G j" i a c ^BUH I O'O 69 -2 i8o'o 45 1 180-0 ■M HHH 2 180 *3 - I O 2 2 171 ^HHl I O i73'8 209 2 l8o 2 3 15 2 b^^hss^ki I 354 11 I O 174 2 1 173 ^^■Bl^Bi 2 8 - I - 2 7 20 1 353'3 2 H^^hH^bI 2 O 162 9 O 2 2 3 173 58 Hill = 2D = 2 - 1 - 1 186 j HH» $181] -5 1 O-3O I 6 ■ I KHNflBfl -'4 2 -2 4 3 286 1 1 Hi -3 3 -1 8 2 2 342 IB fill sssass m*m 36 Mr. Ernest W. Brows, Theory of the § V= + o"-ooi C sm{6+f,J + i( J - T) + a), Jupiter. = 2 D-Z = 4D-Z a (7 4 180 4 3 183 20 2 iSo'o 804 1 7 3 260 4 2 3S3 7 4 187 1 3 187 6 2 67 306 1 280 5 2 18 9 1 107 2 e=2i) + i 2 180 3 2 180 1 2 7 1 - 2 180 7 - 1 -2 7 B=2l 3 2 180 *3 1 2 353 #=2?-2D 2 2 iSo'o 187 1 2 i73'3 190 2 2 162 = 2/-4D 2 2 9 1 2 173 6 = 3/ -2D 2 ISO 7 I 2 173 0=2F-2B 5 2 l8o 0=2F + Z-2D 2 2 l8o 2 0= _ 2F + /-2D 020 I 1 o 81 4 311. SV-= +0" -ooi C sm{6+f $ + 1(8 -T) + a°}, Saturn. 90 1: + 198° 2V + 228° 2 D 2D - r + 338° 1 + v + 192 c 24 2 O o 180 90 90 C 3 3 3 312, SV= +o //- ooiC sin >|/-, all planets. ♦ 14 z - r + 168° 4 2D - 1 17 2D - l - V + 168 2 2 D + 1 2 2 D - 21 :2 D-Z i a C - 2 180 14 I - 2 270 4 c 2 39 1 2 2 I^^^^^^H Motion of the Moon. 37 HlM 3*3- w= + o"-ooi s in{0+/r + z(T-V) + a°}, Ferms. fif^^^^^H ■ 6= ±F (9= ±F + Z <9 = «;, §lf^^^^^fl| j i a y * c i * a a ft^^H ^^H O I o 5 010 2 O -7 285 2 if^^^Hi 2 3 2 1 -6 3 H^HI 3 1 3 180 6 -5 ^m i s 90 2 -4 6 l^^Hl 2 - 18 209 2 <9 = F + ?-2D -3 9 fi^^Hl 1 180 2 - 2 1 4 It^^^^BI <9 = F + 2D 2 6 - I 27 i^Hl 0-4 2 3 34 I 105 15 1 ■!! - 3 180 1 4 4 2 6 1 hH -2 3 5 2 3 3 i^^H - 1 3 1 5 90 4 -2 -8 255 2 if^^^^^K^n 6= -F+2D 0-8 180 - 7 - 6 - 5 2 3 9 2 6 18 4 = _F + Z-2D 1 1S0 2 2 6 3 3 2 -7 -6 "5 -4 -3 75'3 — 2 3 i^Hl 9 I^H 25 l^B 72 l^Hl 18 i^HI - 4 2 3 4 4 - 1 1 fl^^HI - 3 o'o 45 S 2 1 6 i^H - 2 - 1 - 1 - 6 90 14 9 2 2 6 18 4 <9 = F-Z- 2 D 2 3 3 l^H - 5 270-0 68 3 'i8o 2 = W]-2D - 4 90 2 1 5 270 4 ~ 2 -3 255 2 H^^BiS e=±F~i 61= ±F+2Z- 2 D 6> = w 1 ±Z Hi 1 2 3 180 3 - 2 - 3 75 ± 4 1 |H 2 1 2 6 18 3 If 3 14. 527 = + o ;/, ooi C sin \J/- } Jupiter and Mars. * C : rf/ C . $ IBlisil ±F + J + 34 8° 2 + F + Z-2D + 2J-2T 36 Wj + J + 69° ^■1 1 ■'' F- 2D + 2J-2T+ 180° 23 ±F + J-2D + 3J-2T+i73° 14 F + 2D 2 ff^^K»-.]| F-2D + 3J-3T 5 ±F + 2 Z- 2 D + 2J-2T+i8o° 8 -F + 2D 1 HI $ - F - 2D + 2 J - 2T 3 ±F + 2Z-2D + 3J-2T+ 173 9 «; 1 - 2 T+75° 8 B^il ±F + Z+ 2 J- 2 T+i8o° 7 «ff 1 + 2M-T + 345° 10 ±F4-Z+ 3 J-2T+35 3 ° 3 ifj + T-J + 81 2 HP 3 8 Mr. Ernest W. Brows, Theory of the 315. § (Parallax) = + o ,/- ooi C cos ^. I - 2D + 3 T - 3V 1-2V+ 2,1 - 2T Z-2D + 30 -2T+173 C 6 7 3 * c i3T-8V+32i° 3 Z + 3 T-ioV + 33° 35* Z+i6T-i8V+i5i°-o 1455* Z4-29T- 26V+ii2°-o 108 I+21 (T-Y) 3° 2D- Z+21T- 2oV + 2 73°-0 126 2D-Z + 8T-i2V + 303° 33 2F-2D + 6I-5V + 270 54 3Z-2D + 24(T-V) 10 (! + 2F-4D-i5(T-V) 2 D+i2T-i5V+ 2 62° 13 D + 25T- 23V + 190° 13 3D-2F+19T- 18V + 272 2 316. Terms added to the elements. §w 1 = + o //- ooi C sin -\J/-. F + 24T-23V + 28S D-Z + F + 2o(T-V)+i66° D + ? -F+I7T-I8V+7S" 3D - 3I + F + 2 5T - 2 2 V + 1 34° 2 D-/ +5 T-4Q + ii3° 2D-Z + T-3Q + 105 2 r-Z + 3T-4Q + 67° 3D-F-Z + 2T-3Q + 47' 4D- 3Z + 25M — 23T + 67° D-F + 2M+165 io q + no" 8w 2 = + 2" - 69 (No. of years from 1850*0) + o"'ii8 sin (1+ 16T- i8V+33i°-o) Sw 3 = - i""42 (No. of years from 1850-0) + i""86 sin (w 3 + 29o° - i) + o"-i72 sin (/+ 16T- 18V+ i5i°'o) 3 2 8 2* 3 75 3 2 4* 17 7 8y= +o"'o83 cos (w 3 + iio't) Motion of the Moon 39 CHAPTER XII. THE INDIRECT ACTION OF THE PLANETS. Section (i). The Disturbing Function. 317. Transformation to coordinates used in the direct action. — The disturbing function for the action of the Sun on the Moon is (Chap. I., § 3) % \ ft 3Z2 + tP 2 cos 2(V-V') + % £ cos 3 (V- V) + f (Tiz^) p cos ( V- V) j- . (1), to a sufficient approximation. Let Sr', SV be the perturbations of /, V from elliptic motion, the plane of reference being the same as before. Put Sr' jr' = Sp' and neoiect powers of Sp', SV 7 beyond the first. Then the disturbing function due to Sp', SV is R = J -^ 4?- b ^8p'{r 2 - 3 z 2 + 3p 2 cos 2(V- V')} +SF'{ 2/0 2 sin 2 (V- V')} •¥{#P 3 cos 3 (F- F) + (r2- 5? 2)p C os (F- V")} + 8 F{fp s sm 3 (V- V) + £(»•* - 5^ sin ( V - 7')}' (2) Replace the functions of the coordinates of the Moon and Sun by the expressions given in Chap. X., § 285, so that R will now denote that part of the disturbing function which depends on the lunar angle 0. We obtain ii = _ 3?» a- a Sp'(M 1 + |i¥ 2 ) cos 6 + 8 V'M S sin 6 - 8p'-,(M 6 + 5 M S ) cos + 8 F'?-,(jJ^ + \» M g ) (3). We can therefore obtain the required lunar functions directly from the results given 111 § 292 if we multiply all the series there tabulated by aljr'. 318. Final form of the disturbing function. — Denote the coefficients of these functions by accented letters when the multiplication a'/r' has been made, so that 6 is now an angle in the products of the series of § 292 by a'/r'. Let an angle in Sp', 8 V be (p, so that we have 8p' = p c cos <f> , 8 V = -y, sin cj> , m' = n' 2 a' s . Putting a x for a/a' (§ 293), we obtain R = i«'%2( _ 3 )[( M ; + |M 2 ') Pc ± M s 'v, + ojCJ/g' + 5 Jf 8 ') Pc ± ^(£4// + V^s'K] cos (0 ± <p) . (4), which is in the required form (§ 279). 40 Mr. Ernest W. Brown, Theory of the For terms p s sin <p in Sp', v e cos <ft in SV, replace p e by ±/> g , ±v s by v e , cos (Q±<p) by sin (0±(p). The values of fy/, 5 1 77 will be taken from Newcomb's tables of the Sun* (with some corrections). He tabulates io 9 log 10 (i + §p') = io 9 log 10 e . Sp'. If p e , p g denote his numbers, the parts in It which depend on these quantities must be multiplied by io" 9 log^io. The coefficients v e , v 8 are expressed in seconds of arc ; 1 shall consider them as expressed in units of o //- ooi, so that the parts of B which depend on v e> v, must be multiplied by io~ 3 /2o6265. The formulas of § 279, Chap. X., will then be available if we put m"/m' '- -j.ict 9 log e io= — [9*83934], so that /= -[4-13292], /= -[1-45682] ..... (5), and multiply v e , v s by the factor io^ s /2o6265 ^ io^ 9 log,. io = [32335] ..... (6). Then A is the portion of (4) within square brackets, after v s has been multiplied by this last factor. Section (ii). The Computation of Sp', SV. 319. Forms of expression. — In this chapter perturbations of the first order relative to the masses of the disturbing bodies are alone retained. If we had used the method of the variation of arbitrary constants to find Sp\ SV, the variations of the six elements of the solar orbit would have been obtained in the form at + /8 + 2& COS (Xt + ix) , where a, /3, h, A, p. are constants ; in the coordinates we have a similar form, with the exception that in the elliptic terms k is of the form k't + h" and a = o in Sp'. Further, we can put a = ,8 = o in SV, All the periodic terms, except those which are indepen- dent of the argument of the disturbing planet, have therefore constant coefficients and are taken care of by the preceding method. Hence we have to consider only the terms Sp = M + 1(p { + tpi) cos (W + a.) , (*=i, 2,...) .... (7). 8 V = 20',; + tv[) sin (W + a.;) , 320. The non-periodic changes of the solar elements. — Now the solar eccentricity is an observational quantity, and we can therefore choose our arbitraries such that i\ = o. The other i\ and p,, p s are then so small that they may be neglected, and all the portions of the coefficients which depend on t may be expressed by a term e\t additional to e'. We have therefore only to add to the previous values of Sp' the term p^ cos (I' + %), which is treated in exactly the same way ; Set', which gives a constant term to R, and which is treated as in § 2 70 of Chap. X, ; and, finally, the effect of a variation of e'. The mean motion of the solar perigee is not quite zero, and therefore dl'/dt is * Amer. Eph. Papers, vol, vi., pt. 1. Motion of the Moon. 4i not quite equal to dT/dt; the only term sensibly affected in the Moon's motion is that with argument V, for which the divisor n' instead of n' — dvs'jdt has been used. The treatment of the variations of e', xs' require special methods ; that of e' produces the well-known secular variations of w lt w % , tv s ; these have been many times computed, and their theoretical values are not in doubt so far as the lunar equations are concerned. It also produces terms of the form at + b in the coefficients of the periodic terms. These might be computed by means of the equations of variations, but I shall, in the next section, give another method which is much more simple for computation. 321. Corrections to Newcomb' s values. — The values used in the solar tables (§ 318) are taken from his memoir* giving the computations. There are two sets of values in the memoir, obtained by independent computations, and the values of Leverrier are also given for comparison. These four sets of values were compared, and those in which the results agreed within the limits of accuracj 7 required were accepted. But certain of the coefficients (1) in which Newcomb and Leverrier did not agree, (2) in which Newcomb' s two sets of computations differed, and Leverrier's results were not given, (3) in which the degree of accuracy was not sufficiently high, or (4) in which the coefficients had not been obtained, have been recomputed. For this purpose the ordinary direct method was used — a method so well known f that it is unnecessary to do more than give the results ; these are included in the tables of Sect, (v) below. Nearly every one of the few errors found was typographical and easily detected. Newcomb has expressed doubts as to the sufficient accuracy of Sa' and the coefficients independent of the planetary arguments, and he has recomputed these portions. J I have thought it worth while also to recompute these parts by a modifica- tion of the direct method, shown in the following section, which gives the required formulas rapidly, instead of following the method of the variation of constants adopted by Newcomb. 322. Computation of Sp' for the portion independent of the planetary argu- ments. — We shall only need terms of the second order with respect to the planetary eccentricities and inclination in the constant term, and terms of the first order in the coefficient of the principal elliptic term. In order to get the former, we do not need the second elliptic term, since it can only produce a non-periodic term in combination with a term of the same argument and therefore one of the fourth order. Dropping accents temporarily, we have for §r, T7 -(r8r) + n' i —r8r dt 2 ' r 3 dR , --a-—- + 2«| da 1 5ff? dt = (A + B cos I + C sin Z>i 2 a 2 / dw-. dw 1 (8). where 11 is the disturbing function of the Earth's motion due to a planet, and I is the Earth's mean anomaly. All the letters except t are supposed to be accented, and A, B, C are quantities whose squares may be neglected. * Amer. Eph. Papers, vol. iii. t See, e.g., Cheyne's Planetary Theory, chap. vii. % Astron. Jour., No. 590. Royal Astron. Soc, Vol. LIX. 6 mm 42 Mr, Ehnbst W. Brown, Theory of the Putting a 3 /?- 3 = 1 + fe* 2 + yi cos I, we obtain by continued approximation for the particular integral corresponding to the terms on the right, '~=A(i-%e*) + %eCt + ±t(B-3Ae)&ml-±tC<ioal .... (9). No arbitraries are necessary, since they will disappear in connection with corresponding arbitraries in §V. 323. The equation for the longitude is ncfi JT^SV- 2 4(»-§r)+ $>= - 2a j'^dt- 3 n j &df- = (D sin l + F cos l + Ft)na? , (10), ^ at at J da J J dw 1 suppose. Substituting for §r, we obtain, amongst others, terms of the form at cos I, fit sin I. These terms can be eliminated by supposing that rs, e receive increments Sm, Se proportional to the time ; as we are not computing these increments they may now be neglected. The constant term only adds to the observed value of the mean longitude for t = o; it may therefore be dropped. Let on, Se, §l be the changes necessary in n, e, I for t = o, in. order that the mean motion and the principal elliptic term may have same form as in undisturbed motion. Then (1 - |e 2 )8 V= t(F - \Be + %A& + hi ~ |e 2 8re) + cos l(E - C + 2eM l} ) + sin l(D + B- ^Ae + 2Se) . The coefficients of t, cos I, sin I, equated to zero, give on, Se, $l a . Finally, substituting in (1 + e 2 + 2 6 cos /) — - + — - Se cos I + eSl () sin I + JeSe , which is the total addition to log r (that is, the required Sp'), we find the terms A(i + e 2 ) + §F(i + %e 2 ) - ^Be - $De + ±(B + D) cos I + 1(0 -J£) sin I , , . (11). Let B = R Q + R c cos I + R s sin l = E + R 1 , and denote by J the operator ad/da. Then „™ .„...,. „., „ JR. .fdlt v?a?(A + Bcosl+0 sin I) = a-t" + 2 "-- dt = IR, + (I+ 2)R, , da J flWj m%%F+ D cosl-F sin I) = - 2a— - 3 l d h —dt = - zIR - (2/+ 3 )B, . da J dw 1 The expression (11) becomes on substitution of these values w«a*[ -^(i-e2)/R + ^(7-1)5,-1(1+1)^ cos Z-K-?+i)B. Bin q. ■ • (12), a simple form which it is easy to compute. 324. Let us now restore the accents and return to the usual notation. Then R Motion of the Moon 43 becomes the m" j A of § 294. The terms required may be taken directly from Levekeuer's expression (§ 298). We have, for outer planets, RJkaa" = A4 ° + |(e' 2 + e^)(AA + AJ) - ly'ViA - U'e(AA +AJ- A A) cos (to' - ts") RjXaa" = - le!AA + \e'(AA - A J) cos (oT' - ct") , Ii s /Wt" = - fe"^ 1 - ^V) sin (tf - xs") , where 1 use the notation of Sect, (ii), Chap. XL, and certain relations* to reduce the expression for E ; also X = m"/'m'. The required formula for §p' becomes, on making use of the relation I(j> p = (p+ 1 )4> p+1 + f>(p p , satisfied by <p = A, (p = B, for outer planets, t S p ' = ,1-^'aa" { - UA + ^(2 A A - Af) - ^±^BA + ly'^BA + e AL(BA + AA- UA + ±AA) cos (W - ST") Til (4 ■■& 4 + r ^ a a" I ^(AA + A^) - e -(AA + } T AA - IAJ) cos (a' - To") 1 cos V + ?Laa" \ 6 ^(AA + ±AA - Un 1 ) sin (T3 1 - rs") 1 sin I' . mi 2 ) For inner planets, E only requires the change a' for aa", but jR c , i?, are given by R e jXa' = l(^o + i«) - --(2^1 + AA) cos (gt' - ct") , R s /Xa' = ^{2 A A- + AA) sin (rs' - or") , 22 2 while in the expression (12) for Sp' we put — /— 1 for I. The values of a"A l p , a"B^, a' ' A' 9 , a'B l p are given in the auxiliary tables for the direct action. \ Those of the other quantities are found in Sect, (iv), Chap. XL The final results are included with the other terms in Sp', SV. Section (iii). Second Method, Application to Non-periodic Changes. 325. Statement and Solution. § — The method may be regarded as a particular case of the genera] problem of four bodies, or of three bodies, or as a general method for treating any motion which is transmitted through one body to another, according to the view we wish to adopt. The last view will be that most convenient for our immediate purposes. Suppose that we have been able to solve, in terms of t and arbitrary constants, a dynamical problem which has a force function F. This function, expressed initially in terms of the coordinates, may also contain t explicitly and given constants. I shall suppose that it contains t explicitly only through certain functions of the given constants, u h , some of which may therefore be constant and some variable. Now suppose that, owing to some external agency, the u h are not the complete values of these * A.P.E., sect. iii. t The formulae do not quite agree with those of Nbwcomb given in Astr. Jour., No. 590, but the .umerical results agree with his as given in his paper, Cam. Inst. Publ., No. 72, p. 90. % A.P.E., sect. v. § I have given the method in a paper in the Tram. Amer. Math. Soc, vol. vi. pp. 332-343. n WWWJW I WW I WI^ I li 44 Mr. Ernest W, Brown, Theory of the given functions, but require certain additions, Su h , whose values in terms of the time are given. The ordinary method of treatment consists in substituting these new values in F and obtaining a disturbing function * 2 h (Su h dF/du h ) ; this is accounted for by find- ing what variable values must be given to the arbitraries, so that when these values are substituted instead of the constant values in the expressions for the coordinates and velocities, we shall have the complete solution of the problem. In this method no account is taken of the fact that F retains the same form with respect to the u k what- ever values may be given to these functions. In the memoir referred to, an attempt was made to utilise the absence of change in the form of F by considering the problem in the following way : — To find the variations of the arbitrary constants when not only their variations, but also those of the u h , are substituted in the expressions for the coordinates and velocities. I proved that if this plan were followed, the solution was equivalent to adding a disturbing function B-W^-tg), ...... ( I3 ), where the U h are defined by the differential equations dU h _ oF _, T1 3 du k dB . ~tj ■ — 5 ■ 2 <k u k-^ — —;r - t. — j • • • • • ■ (14), at du h ou h dt ou h v ^' it being supposed that F is expressed in terms of the coordinates and the u h , that dujdt is expressible in terms of the u h , and that B is expressed as a function of the constants and of those u h which are independent 1 of t. 326. Application to the secular changes of e', ts'. — We have initially e' , v>' con- stant. Let Sef = e{e't, Sxa' == vs^t. The u h are n', n't + e', ts\ e\ and therefore rs\ e ' are independent of the other u h and of dujdt. Also (loc. cit.) (J e ,, U w ' contain no non- periodic terms. Hence p '('OF,, ^.,[dF-,, S=- ei je (It-W, dt ...... (IS) , oe J oXS in which the non-periodic term arising from dFjde' must be -dropped. If we substitute this value of R in the equations of Chap. X., as we have no non-periodic part of R, §c ( = o, hv H = o, and therefore the secular accelerations are obtained by putting e'(i +e 1 't) for e' in the values of c v c 2 , c 3 expressed as functions of n, e, y, e', n', and finding the values of n, e, y which result, t The motion of rs' produces nothing in this connection, since it is not present in the c { . I have shown J that if we neglect af , a quantity which is quite insensible, the variations of n, e, y can also be obtained from the equations e.db„ , *db„ „ <> d f u \ c 2 d^ + c 3 8— ? = fS— (i- , ai = n,e,y (jd) * For simplicity, only variations of the first order are retained, but the methods are applicable when we take in higher powers. t This is Newcomb's theorem, Amur. Eph. Papers, vol. v., pt. 3, p. 191, % Proa. Lond. Math. Soc, vol. xxviii. p. 154. Motion of the Moon. 45 where (f*/r) denotes the non-periodic of fi/r , the functions being expressed in terms of n , e , y , e', n', the first four only receiving variations, that of e' being given. 327. There remain the periodic terms of R. Since e\ vs' occur in R only through r', V, we have only to put the periodic terms Mr de ■a, -: dp drs" SV-- . ,dV „,dV de dvs' ' ■ (17) in the formulae of §317, and, after rejecting all non-periodic terms, integrate; the resulting disturbing function consists only of periodic terms. The variations of the elements are then substituted in the coordinates. In accordance with the principle of the method, we must also put e'(i +e^t), m' + ra^t for e' vs' in order to obtain the true values of the coordinates. '- ; 7 No other secular terms can be produced from the secular variations of the solar elements. It will be shown in Sect, (iv) that those of the inclination and node only produce periodic variations. The method of this section might have been used for all the indirect perturbations. Section (iv). The Motion of the Ecliptic. 328. Choice of the Mean Ecliptic. — Owing to the action of the planets on the Earth, the plane of the Earth's orbit is not fixed, but has a motion which can be ex- pressed as secular and periodic variations of the inclination and longitude of the node with reference to some fixed plane. I choose as fixed plane the ecliptic at the date i850"o, and refer the motion of the Moon to the mean ecliptic at time t* The periodic perturbations are then included in the terms of the Sun's disturbing function which depend on z', portions which have been previously neglected (Chap. I., § 3). To be iu eluded in this mean ecliptic are one or two minute inequalities of very long period which then give rise to no terms in the Moon's motion, but which would do so if included in z'. 329. The disturbing function for the moving ecliptic. — This is most easily found in a general manner. Let x, y, z, u, v, to be the coordinates and velocities of a particle of mass m, referred to rectangular axes which have velocities 6 1: 6 2 , 6 3 about themselves, and let the force function be denoted by mF. The equations of motion of m are then given by du_ oil dv_ dH (ho _oH lit ox dl~ dy ~dt~ 9.3 dx dH dy _ dH dz m lit"' dll dt ov w ow where H=-^ + v 2 + te 2 )-F-B ! R = vz0 3 - wx6 2 + wy6 1 - uy6 s + uz6 2 - vz6 x , if we assume that 6 U 6. 2 , 6 S are independent of x, y, z, u, v, iv. * I have discussed this point in vol. lxviii. pp. 450-455, of the Monthly Notices, and have also given there the substance of this section. 46 Mr. Ernest W. Brown, Theory of the When Ii = o, the equations become the same as those referred to fixed axes, and therefore li is the disturbing function for the motion of the axes. Let i! be the inclination of the moving ecliptic (xy plane) to that of 1850'Ci, t the longitude of the node on the fixed ecliptic, and for the origin of longitudes on the moving ecliptic take the "departure point" whose distance from the node on the moving ecliptic is the same as the distance from the node of the origin of reckoning on the fixed ecliptic. Then by Euler's equations „ di' dr ■ ., ■ n di . dr ■ „ a dr , ., N , „. 1= = COST- SHI I Sltl T, P 2 = S1I1T + — sin* COST, 0, = —-(cos& - I) . (18). at dt at at at The values of i', di'/dt, dr/dt are small quantities of the first order, so that their squares and products may be neglected. On substituting the values, so limited, in li, we find the factor di'/dt common to all the terms, and therefore, since u, v, w differ from dxldt, dyjdt, dzjdt by quantities of the same order, di' r n~( dz /l!j\ ( dz dx di r , ( dz di/\ ( dz dx\ . , , tA = (y-7;-Z-T- COST- [X-—-Z-— suit .... (10). \ dt dtj \ dt dtj v Jl This disturbing function is available for any moving ecliptic so long as we may neglect the squares of its perturbations, but under the assumptions of § 328 we substitute for i! only its secular part. The resulting disturbing function I denote by R t . 330. The disturbing Junction for 'perturbations of the Earth out of the plane of reference is, if we neglect squares of z' and the terras dependent on a, E = -—:-^- ^d ...... (20), r " r * by § 3, Chap. I. With the notation and limitations of the previous section, z' = i'(l/' cos t - x sin t) . ... (21), Since this expression has the small factor i ', we consider t as a constant. In the paper referred to in § 328, I have shown that the expression (20) for li can be transformed into „ .,\( d % u d?z\ ( d?x ddz\ , \ ., dQ , N R = t 1 \z~- - y -^ cos t - \z~^ - x—; sm r > = - 1 ~^~ , . - (22), I V df df J \ dt 1 dt 2 J j dt ' which again is a perfectly general expression for the disturbing function when we can neglect squares of the perturbations. The value of Q is that given in the previous paragraph, and therefore the computation of one function serves for both disturbances. Under the assumptions of § 328 w r e substitute for i' in (22) only its periodic part, neglecting the minute perturbations of i' which are of long period relatively to that of the Moon's node. The disturbing function thus limited will be denoted by P H . i i Motion of the Moon. 331. Computation of Q. — We h ave Q = ^psm(V-T)-z^L{p 8 m(V~r)} 47 = real part of (n-n')[Dz ■ wg-'e 4 "! 1 - zD(u£- 1 e w v)]e- " awg- 1 ■ D(a)-(D+i+m)(iz -u£fy k(i + m)a 2 e( w i - T )' , The expansion of the portion in square brackets has been given in § 291, Chap. X. Let .1/ exp. 9 x i be one of the terms of this expansion. Then one of the terms of Q is given by Q = 11AM sin (6 X + w l - t) = nAM cos ($ 1 + s 332. Computation of R v — Let di'jdt=p . Then n ' m pi'VM cos [6 1 + w 1 -t- 7T\ V. and -therefore if we put m" 'jm! for the factor outside the square brackets the equations of § 279, Chap. X., can be immediately applied ; here A = M. The value of p is* d'-^jio per annum, n'=3548 //- 2 per diem, and therefore 4('47io)k r- n giving m (36s'25)(3548-2)m, /=[6' 2 333L /' = [i'SS72]- In finding derivatives with respect to n, we must use d(nka, 2 A)/dn. 333. Computation of R t . — Let P cos (p be any term of i' reckoned in units of o /,- ooi, and s x the number of seconds in the daily motion of the angle Q l + w l — t = 6. Then, using the first form for Q, we have where 7? =fLi m t' 2 -d 2 A cos (6 ± <f>) A=- ^MP , k , r— m' 1000 206265 3S48'2 m — - = +[12-5149] . The computation of derivatives with respect to n must be made with SjAfa 2 . The last fact makes, the computation easier with formulas (20), (21). We have B 2 = fn'H'^J pe {sm(V-r)- sin (V- 2V + r)} = \riW-, { - M'P cos (0 ± <£} with h" = t , + |w'% 2 ^{ Jf 4 "P cos (6 ± 4,)} with r = 2F-r, using the third formula of § 285, Chap. X., after multiplication by a'jr'. For these * Leveebibr, Ann. Ohs. Paris, vol. iv. p. 50, after correction for the adopted masses of the planets. 48 Mr. Ernest W. Brown, Theory of the terms it is sufficiently accurate to neglect the solar eccentricity in finding the Moon factors ; this gives M\ = M'\ — M A , F ; = T. The value of m"jm' is K= ?- .—^=[5-16273. m 1000 206265 The terms, arising from these disturbing functions, in the results at the end of this chapter, are those in longitude which contain the argument w s explicitly, and those in latitude which contain the argument w 1 explicitly. The most important are two of the latter with arguments Wjdt^T — 3Y) and having coefficients o"'OJJ and o //- 030 respectively. Section (v). Numerical Values of the Earth's Perturbations. 334. Sources. — The general values for op', §¥' are taken from Newcomb's tables of the Sun (§ 318), with some corrections and additions (§ 321) ; the secular variations of e', ra' are from the same source. The terms independent of the planet's arguments are found in § 324. The values of i', r are from Leverrier,* after correction for the masses of the planets adopted here ; the coefficient of 2J — 2t in i' was recomputed. 335. Notation. — The values of §p', SV are given by Newcomb as cosines and sines of the mean anomalies ; it was convenient to retain them in this form. Leverrier gives the values of i' in terms of literal arguments ; these were combined and expressed in terms of the mean longitudes, which is the final form for all the inequalities due to planetary action (Chap. XV.). Hence, I put $, = £ cos {f(T - rf) +j"(T - vs")} + £ sin {/(T - To') +i"(T" - tr/')} , 1 = ! c cos (j'T +/T" + «') . Also, v e , v g are expressed in units of o""OOi, while p a p s are expressed in absolute units multiplied by io 9 Jog 10 e, that is, they are the coefficients in io 9 log 10 (i +Sp). For the secular terms, de' jolt = e/e', d&'/dt — vsr 1 . * Ann. Ohs. Paris, vol. iv. p. 50. • Motion of the Moon. 49 336. Venus, T" = V. ,/ o I 2 O I 2 3 o 1 3 4 2 3 4 5 6 3 4 5 6 7 5 6 7 8 6 7 8 9 12 J 3 14 16 1 8 Pc P* Vs Ve + 627 + 14 - 7 - 85 - 39 - 67 + 33 - 2062 _ 1 1 46 ~ 4228 + 2353 + 68 _ 14 - 34 - 65 + 14 - 8 _ 8 - 3 + 4 + i - 3 + 84 .!_ 136 + 60 .. 99 + 3593 + 5822 + 2903 - 4702 - 59 6 - 632 - *737 + 1795 + 40 + 33 - 33 + 3° + 21 + 1 - 13 + 44 + IO44 + 27 - 666 - 381 - 1448 - 397 + 1508 + 126 + 148 - 684 + 763 + 14 + 13 - 12 + 12 + 6 _ 1 - - 166 + 337 - 93 _ 188 - 5i + 189 - 38 - J 39 - 25 - 9 1 - 42 + 146 + 3 + 5 - 4 + 5 - 134 + 93 - 69 - 47 - 39 + 43 - 2 5 - 28 - 37 + 136 - 33 - 119 + 0-3 - 27-9 - 1 + 154 - 80 + 8 - 38 - 4 - 24 ~U 7 - 13 - 4 10 -f 10 - 7 - 6 + 3 - 12 + 3 + 14 - 38 - 17 - 18 + 8 - 14 - 19 _ 7 + 9 - 43" 2 + 8-i - 41 - 8 + 9'oo - 7-71 + 416 + 1251 - 25-2 + 22-3 + 24 + 21 24 24 3° 43 39 209 180 209 209 209'4 29 299 see § 350. Royal Astron. Soc, Vol LIX. rnmmm 5° Mr. Ernest W, Brown, Theory of the 337. Mars, T" = M. o I - 2 - I O - 3 - 2 - 1 o - 3 - 2 - 4 - 3 - 2 - 1 - 4 - 3 - 5 - 4 - 3 - 5 - 4 - 6 - 5 - 4 - 6 - 5 - 7 - 6 - 7 - 6 - 8 _ 7 - 9 - 8 - 9 10 11 13 15 i7 Pc fs Va - 13 + 5 — 3 5 + 6 4 - 92 + 119 - 167 + 27 - 6 - 47 - 13 5° 10 - 573 [976 - 567 4- 64 - i37 - 617 - 18 - 25 + 15 - 154 - 67 - 118 - 77 - 201 - 153 + 46 - 17 + 3 2 + 461 + 125 + 483 + 43 + 96 - 256 + 6 + 8 5 + 87 - 62 + 69 + 87 + 17 + 200 3 + 30 2 - 102 + 94 - 113 - 27 - 4 + 100 + 4 + 60 + 3 - 26 + 28 - 72 12 - 9 8 8 - 44 - 10 + S _ 6 — 12 - 3° - 16 - 25 4 - 17 I 5 + 7 _ 3 + 5 + 14 + 6 + 18 + 17 - 10 + 15 + 8 + 3 + 42 1 + !5 — 1 4 + 3 - 33 - 17 - 14 - 16 0-90 - 5-92 - 3* 1 '3 - o-6 + 24 - 216 The last figure was not computed, and is + 4c + J 963 — ] 659 - 24 + 53 + 396 + 11 - 131 + 526 + 7 + 49 - 38 _ 20 - 104 _ 11 _ 49 - 78 + 6 + 5i - 17 + 13 + 60 + 2 - 7 + 9 - 12 - 13 - 3° + 13 4- 20* _ 10 not needed 263 Hiiilii Mm;,,,-, .,/' i i,,' M.„,„. 5' 338. Jupiter, T" = J. Jilllil§ll Hilt Pc ' f* ■ ' : • "' .•• ■ *s ''/'■■ ' ''■■'.' 'Vc, .';•• . :■•';.. it- : <; •iJ: ; 3 •;■••; - ; 513 IllSlllllfl #111^18%^ 39 ■ ■ +' * :; ?2 •"'.';• ~ 5 1 *V*V ; ' •- '2 •••';■• + 5 — I 3 _ -> - 78 + 193 - 52 ■'. ' : ~ >55 " ; illilllllli ;■•:>'. ■ >.;+'* 56 + 7067 + 59 - 7208 e -.- ; ■ : • + .22-7'. / - 89 -2582 - 3&7 - 6 iltlltlllfl + 79 ' : + 9 ''=-■• 73 . .: '. +.-'.'8 , ', lltifill 8111 + .lb. a: \:;. - 17 + 68 - '+ -i 1 • • - _• . +.4021 •..•'• - 203 :-±- 21<Z&:<: + 136-. .- '3'3 Slllllllll 1 > +' [376 + 486 ' +IS18 ' '•'• -' -537/ . ^llllllll^llii - I 8 . - -70... — 22 'S 2 Pllfll 3 .;■ V/ : .' . -4" 43 ; + 278 • '+.' 27" ; - 162 J M"-': '":''-¥ 796 — 104 + 55' "'.'.+ . 7i lllllillllt ''■'■ ■■''.- '.*' : * 172 • + 26 , + ••2.08 - 3' ... 18 - 4 4 ^'."•"''■ : ' ; ~- 29 + • .5 - 16 3 - 3 7.; ;- ; -V : -*' 13 • +• 73, V + 9 . - "43 '. - 2 ; -" : .- * : '.' ; + no - iA- ■ ■+ v 78 ' ■ - 17 - 1 ,> : 17 + • 1'.. ' + 23 - 1 354 1 1.) 2 I 5° ^39. Saturn, T" = S. p- \ '■•• . ■""'.: 24 2 1 '/'■':.■ + . J 5 . '• ' +•;• .; :,,*' II - 3 34i- .-!// f/fililrls I .4 422 : ' ;'+'. 79 "•■•'+• 412 - 77 Secular terms. O ' : V. 7-89 • : ~" O' 53 - 320 - 3 1 ~ ' -'+' ; 8 : ° - 8 : ; S'tt'j' = ; p- '..' - 2/ W-:-/-^-:' 1 5 ^ W.'SSv'i: 57 - 101 '••/+•' 38 sr/ = +[6 "9537]"' 1 - ""^ 103 ■ ■ '—y 44 ■ ; ~! 103 • + 45 S» — o'.. ; . '+'• 0-31 ' ' — • - 0: 56 - *7 . + -2 «]' --P "59.68]«' .; 3 :-.>"- ' ■ 30 ' ; \ ; ^:j^-i 11 ■^-r::"^ 20 •.• +:, 7 T = 173° •46 1 — 16 /■vrr--^^ 6 ^; : x^ 16 + 6 "f --- +[7-5604].// 340. Mercury, T" = Q. 27 5 2 + - '3 19 Mr. Erxest AY. Brown, Theory of the Section (vi). A Sieve fere the Rejection of Insensible Coefficients. 342. Terms to be considered. — Just as with the direct inequalities (Sect, (iii), Chap. XL), we only need to consider the possible terms of long period. Also, as before, I consider separately those that do or do not contain v: x in their arguments. The latter can be at once dealt with ; the number of possible terms in the lunar factors is practically limited to those with arguments 2D — 2I, 2D — 2F, 2 F — 2 /, I) — I, and in any case it is a brief matter to consider the possible combinations with the planetary arguments of all but 2D — 2/, 2D — 2F, which arc computed in the regular course with the short-period terms. Those that contain iv 1 in their arguments can only produce sensible terms in combination with planetary factors having nearly the same period, that is, a month. Hence we have only to consider the values of coefficients in op', SV belonging to terms which hare periods of about a month or less. 343. Construction of the Sieve for terms containing u\. — The equation for cV is (§ 322) d- ,'«,-., n-o'P , s , .flR' , , fdR' 7 , , -jror) + ——rar==a--^+2nl--—c/t (21), r ■• da J d t ' " dP where R! = rn'jA ; the other portion of M' gives coefficients which are quite insensible in the class of terms considered here. Let q be the mean motion of a short-period argument in R', and therefore in Sr : for the periods approximating to a month in eV we have rlq = m approximately. As far as the effect of the periods is concerned, the four terms of (23) are of relative orders <r, n n , n' t , 211'q. And further, the largest terms in R' with periods approximating to a mouth must have high multiples of T — T" in their arguments, and in this case the two derivatives of R' are of the same order of magnitude. Hence the order of So' for such terms is given bv 11 -a 'bp , d.R' rP ■a . -,- _ - or ila c q- da The equation for SV may be written , , ,- — ,- a at ,(SV): n'ci", ,. , a u ,dr 2 ---'W^ br) ~ad^dU {24). dli' irF dt (25). Similar reasoning shows that the order of the right-hand member is the order of its last term, so that, on integration, SV is seen to be of the same order of magnitude as Sp'. Now A 2 = r'' i + r' 2 — 2r'r"<r, where a- is the cosine of the anode between r', r". Hence .f'lTl . fill ../ -1-"^ ~~ ■■}•'' ^ i \ r m'' r/' z . . (26). "'iliii' ~"dr -— = order of for the worst case, that of Venus. Hence 3^// is of order (iii"ii)i')(a' : PA H )m Motion of the Moon. DO Take first the inequalities depending mainly on Sf. A comparison of the disturbing functions in §§ 294, 517 shows that the order for the indirect terms is to that for the direct terms as $Sp : (m/ / fm')(«/ 3 l& s ), that is, m 2 : 1, since we mav take 3V/ A 6 as having the same order at the worst as i/A 3 - Hence any term which is shown by calculation to have a coefficient less than \" due to direct action will not be sensible in the indirect action. There is only one term left, that with argument 1 + 16T- 18 V, coefficient i4 /; '55 ; the order of the coefficient for the indirect action is this number multiplied by m\ that is, the order o"'oo. (the computed coefficient found below is o //, o6). The terms due to SV are treated in exactly the same way and give similar results. The direct inequalities are all small, and there are no sensible ones arisino- from the indirect action. Hence, there are no sensible terms of long period containing the argument tv 1 mid arising from indirect action in the plane of reference, except a small term having the argument of the great Venus inequality. 344. Terms arising from i'. — The principal argument in the Moon factors is m>, and the combinations of this with the comparatively few terms in 1' which are sensible are first studied ; then the terms of one order higher with respect to the lunar eccentricity and inclination, and so on. It soon becomes quite obvious that the only terms beyond those of lowest order in the Moon factors must be of long period relatively to that of the Moon's node. As the terms in i/A to be considered must have the factor y !,i , their number is very limited. The methodical search for long-period terms was simple. The tables formed for the periods in the sieve for the direct terms were available.* As w B is itself of lone period, it was only necessary to combine w z with terms in that table which contained multiples of w. 2 — w s , w 3 which were either both even or both odd, that is, terms for which the multiples of w. A or F were even. The combinations of these with w 3 were those required. Only two survived, apart from those with the sole lunar argument iv s , namely, the arguments 2W 9 + ?<> 3 - 2.J = W - w B ■ 2D - 2I- w 3 - 2 J + 2T, period 277 years ; T, period 540 years. 2,J = 2F-D-Z+2i- 3 -2j lire latter is quite small compared with the former, on account of the lunar characteristic k"a,e as compared with e 2 k, and the planetary factor is also much smaller ; the period is twice as long, but this only multiplies the relative coefficient by 4. In a note at the end of a paper lately published, t I gave a value for the coefficient of the former term as o"-2\. Since the paper was published the term ims been recomputed with the disturbing function (20) ; this revealed an error in trie former computation, and the coefficient appears to be d'-oo^. It is therefore not retained in the final results. A.P.E., sect, v. Monthly Notices, vol. lxviii. p. 170. mwm 54 Mr. Ernest W. Brown, Theory of the Section (vii). Computation of the Lunar Perturbations. 345. The disturbing functions are given in !$§ 318, 324. 327, 332, 333, the values of the planetary factors in the last section, and those of the lunar factors in Sect, (v) of Chap. X. The method of arrangement was to take one lunar argument with all multiples of T — w' and form the products for all the planetary arguments. The equations of variations have been so arranged that the process of finding the values of hvi, on, Sc 2 , Sc 3 from the disturbing functions is very brief and simple, inasmuch as it was rarely necessary to compute more than two of these six variations. In fact, in the few cases where more than two were needed, a simple ratio, the same for all terms, generally sufficed ; such a ratio was also sufficient in the majority of cases to find all the other variations after the principal one had been obtained. The experience gained in computing the direct inequalities suggested that the work could be much abbreviated by considering the peculiarities of each lunar argument, and these peculiarities are set forth in the following paragraphs. 346. The primaries independent of the lunar angles. — Here §n = Sc z = Sc 3 — o, and Stv 1 is first computed, and then - eSw 2 , so that the secondary arising from the sub- stitution of hv. 2 in the principal elliptic term, - 2e cos I. Sw 2 , is obtained directly. For $w s , it was sufficiently accurate to treat §w s /8w 2 as a constant which is the same for all terms, and indeed for the small terms, kv 2 : Sw 3 : hv 1 are constant ratios with sufficient accuracy. When these primaries and secondaries have been found, the remaining secondaries can be written down almost by inspection, I therefore only give the coefficients of the primaries arguments \l, and of the secondaries arguments \1/±Z. The primaries of very long period arising from terms of very long period in $p', §V are treated in § 352 below. 347. Primaries containing w x and independent ofu\, w 3 .— Here the periods of all the terms are very nearly the same, Sw 2 is nearly equal to Sw t , and Sw s is about }3w 2 . Hence the secondaries are all very small, the largest being less than o /; -oio. All of them greater than o"'ooi were computed, and will be included in the final results, but it is unnecessary to print in detail any but the primaries. The principal variation is again Sw v 348. Primaries containing w 1} w 2 or w z only. — These are the terms in which the secondaries are generally much more important than the primaries, and in which the theorem of § 282, Chap. X., has its full force, the principal variations being $w, 2 , Sc 2 . If the primary contain w. z in the form il + <p (i positive), the principal secondary is that with argument (i -j)l + (j>, and this was first computed in all cases ; then the value of Sw 1 ; §w B was insensible, or only produced very small terms. The terms divide into two classes according as their periods approximate to a month and less, or are much longer. The first class includes the lunar arguments J, 2D -I, 2D + I, 4D-Z, 2/, Motion of the Moon. 53 for which the principal terms, those in longitude, have the lunar arguments o, 2D, 2 D, 4 D, I, respectively. The second class contains only one argument, 2D- 2I, for which the principal term in longitude has the argument 2D — I. I give Sw t , that is, the primary, and the coefficients of the principal set of terms in longitude only ; from these all others can be obtained immediately. 349. Primaries containing iv x , w s or w s only. — The principal variations being Sw 3 , Sc s , the largest term is always the principal secondary in latitude, and the statements of the previous section can be repeated. But only two lunar arguments have to be con- sidered : 2F, which belongs to the first class, and 2F-2D, which belongs to the second class. The principal terms in latitude contain the lunar arguments F, F - 2D, and the coefficients of these four terms are alone set down. 350. Special terms. — -Beyond the inequalities mentioned above, there are two with arguments D-Z-4T + 3V, Z + 16T-18V, and with periods of 94 years and 273 years. The former is computed in the same manner as the other terms. For the latter we require to find §p', SV for the argument 16T- 18V or x6(T-ts') - i8(V -vs"). It was known (§ 343) that the final coefficient was of the order of o //- i. It was therefore sufficient to find the terms with multiples 16, 17, 18 of T — tff' combined with i8(V -m"), so that they might be combined with the lunar arguments I, l-(Y-xa'), l—2(T-xs') in order to give the required argument. The direct method of computation gave (in absolute units), with sufficient accuracy, 8p' •0000059 00s i6(T - &') - i8(V - ui") + 129° 8V - '0000046 sin ( "hence arises the only sensible term, §•«/'!= - o"'o6 sin (1+ 16T- 18V+ 150° f m m (27)- 351. Omitted terms.— The portions of the disturbing function independent of the planet's mean motion are the constant term and those with argument V . The former chiefly produce small changes in the mean motions of the perigee and node, and also slight changes in n, e, y which affect the coefficients of the evection and variation, but the latter are quite insensible. The variation of e' produces the secular accelerations. The terms due to the motion of the ecliptic and the latitude of the Sun do not combine with any others due to Sp', §V. The detailed results given in the next section do not contain these terms, nor the terms in the two following paragraphs ; they will all be given in the final results due to indirect action contained in the last section of this chapter. 352. Terms left as perturbations of the elements. — A few of the primaries which have very long periods and which are independent of the lunar angles are ■m 56 Mr. Ernest W, Brown, Theory of the so left, and their values are to be found with the collected results at the end of this chapter. The principal inequality due to the motion of the mean ecliptic is also treated in this way. But there are short-period primaries due to long-period terms in Sp', SV which are sensible. I shall show that these may be accounted for by the substitution of w-l 4- long-period terms in SV for w{ in the final expressions for the coordinates of the Moon. Consider how such terms would be treated by the second method in Sect. (iii). The chief perturbations of the solar coordinates arise from Sn' and Sw x ' = jSnfdt + Se. Hence we should have Su x = Sit', Su 2 — Sip/, so that d 5, *du, d * ' d <* , ^dw,' rl,-, , — ou, — d — l = on , -610, — o — l -=--oe. dt ' dt dt dt x at at Now- hi', Se only contain the first power of the period as a large multiplier, and therefore the disturbing function does not contain it at all.* Hence all the terms produced by the use of this disturbing function are quite insensible. We therefore proceed as with the secular accelerations, obtaining the variations of n, e, y by solving the equations csc t - = o considered as functions of n, e, y, n' in which Sn' is known. Then §w 1 — \hidt, etc. But this simply gives the primaries independent of the lunar angles, together with the secondaries arising from them, and these we have already obtained. The method therefore proves that for all other terms arising from long-period terms in SV, Sp' w T e are to simply substitute the disturbed values of ,rt', w-[ in the final expressions for the Moon's coordinates. The substitution of Sn' gives nothing sensible. Hence the statement. 353. Computation of the secular variations. — The values of Sn, Sb 2 , Sb s have been obtained in an earlier memoir. + With the adopted value of Se', namely, e/=-[6-5968]< I find 8n= + [Ti'o294.]rm't , S& 2 = - [7i - 8476]mm'£ , 8b 3 = +[Ji'o'j46]nn't , giving 8wj = + 5"-8 242 , 8iv 2 = - &"•&* , Sw 8 = + 6"- 4 6^ , where t c is the number of centuries from 1 850*0. To obtain the values of Se, Sy, we have o = 8c„ = ?p Sn + c iS Se + *p Sy + d ff8e , an de dy de with a similar equation for c 3 . As Sn is known, the values of Se, Sy can be found from the formulae (39) of Chap. X.. by putting for Sa 2 , Sc s the expressions idc„ ,, ide<, ,, - e df e ^' ^di e ^- We find that Se/e. Sy/y are less than io~H C) and therefore quite insensible. * This is true in general. See Trans. Amer. Math. Soc, vol. vi. p. 341. t Monthly Notices, vol. lvii. pp. 342-349. Motion of the Moon. 57 354. Description of the tables. — The portions of the detailed results selected for printing have been described in the preceding paragraphs ; it is understood that all portions greater than o"*ooi have been computed and included in the collected results o-jven in Section (viii) below. The heading to each page sufficiently describes the contents in general It will be noticed that \j/ + a° always denotes the argument of the primary, \f^±/ + a° being the argument of the principal secondary in longitude, and ^±F + a° that in latitude. The angle rs" is the longitude of perihelion of the particular planet considered, and V the mean anomaly of the Earth's orbit. a is the same for all values of C in a given line until a new value is The angle set down. Royal Astron. Soc, Vol. LIX. 58 Mr. Ernest W. Brown, Theory of the 3 5 5 . S V = + o"-oo i C sin { i'V + i"( V - ro") + + a} = + o"-oo i sin (^ + a), Venus. e = o a= 2 D 6i= - 2JJ * *:W + ■ \ * i' i" a G a C a c a c o - I 334 8 33 2 2 152 10 i 330-9 354 33°*9 72 150-9 III 150-9 50 2 48 12 90 3 319 2 175 4 3 28 1 i - 2 122 13 122 5 302 I? 2 1 2 1 '6 5" 121-5 104 3°i'5 179 3° 2 11 3 3i5'4 275 3 2 4 39 i33'7 54 134 30 4 3 2 3 5 136 3 2 -3 91 1 272 2 3 g2'o 50 91 10 272 23 91 ; 4 284-8 158 285 S 2 104-7 46 104 8 5 3 r 4'5 129 322 22 i3 2 14 I3 1 20 6 315 2 133 2 4 -4 64 14 64 3 . 244 9 S 74 13 73 3 255 5 6 287 17 288 3 106 4 106 1 5 -5 35 5 215 4 6 48 3 226 2 7 74 12 74 3 253 4 8 271 19 274 4 90 3 90 2 6 -6 6 2 186 2 9 256 2 7 -7 156 1 Motion of the Moon. 59 §V= +o n -ooiCsm{i , l , + i"(Y-W) + e + a°}= +o"-ooiCsm(^ + a°). ) = l i 2 1 l ■■ ■• 4 -4 a V a 332 1 33°'9 13 296 122 2 121-5 24 335 5 -&■ 91 285 3°9 3 6 2 O CS 64 1 73 287 35 74 271 4 47 1 7 92 18 13 21 5 5 3 2 2 2 1 -I $ + 1 33i 5 302 3i4 3*7 2 73 103 312 244 21 7 1 4 3 7 i <9 = 2D + Z (9 = -2D -1 6» = 4D-^ <?= - 4 D + 2 * i|/-J * t+i i/» ty + l + -Z t? i' a c c c c C G (7 - 1 !5 2 2 I 151 4 20 2 10 2 5 2 I - 2 302 3 1 2 301 6 32 2 2 7 3 *33 2 10 1 6 2 I 3 ~3 271 4 4 105 2 8 1 2 S x 33 3 4 4 -4 244 2 6o Mr. Ernest W. Brown, Theory of the SV= + o"-ooiCain{i'l' + i"(V -w") + + a 3 } = + o"-ooi Csin (| + «1. 6=2l)-l 0= -2D + Z * * + z <l> ♦ -* i' i" a c c a c c - i 332 4 19 i 330-9 41 214 330-9 16 91 2 300 2 358 2 9 I - 2 121 7 32 2 121-4 72 359 122 5 13 3 313*3 !9 100 3*3'3 9 55 4 316 1 6 2 -3 92 S 3 91-0 10 47 272 3 4 284-7 19 91 284 2 J 3 S 3 12 5 26 311 7 37 6 3i3 3 4 -4 64 4 20 244 2 5 74 2 10 6 286 1 7 286 2 S -s 35 2 9 6 S 2 4 7 74 2 8 8 271 1 5 271 4 6 -6 6 1 5 7 18 1 7 -7 336 3 8 -8 3°7 2 0= = 2l 0= -2? i' i" a (7 1 — 1 151 3 I 2 - 2 3 02 5 3 134 1 3 -3 272 1 Motion of the Moon. 61 S V, SU=+o"-ooiC sin {i'V + i'"( V - ra") + 6 + a] = + o"-oo i C sin (^ + «°). :2 Z- 2 D o i 2 I 2 3 4 3 4 5 6 4 5 6 5 6 7 8 8 -4 = - 2l+ 2D * *-z * i^ + Z a c c a c c 159 4 33 2 5 33°'9 3 77 33°"9 4 87 2 3 108 1 122 122 6 301 6 IO 121-5 6 112 3i4 2 36 3 r 4 2 48 3*7 2 92 - S -3 59 91 12 284 11 285 2 33 311 19 312 16 3i3 1 64 4 64 4 2 54 3 74 3 286 2 286 3 36 1 35 1 45 1 74 2 270 2 270 3 236 4 - I - 2 -3 331 122 3 r 4 284 311 )=2F-2D c I I C 11 6 33i 121 3i3 gi 285 313 -2F4-2D * * + F C I 11 2 14 7 1 4 2 Mr. Ernest W, Brown, Theory of the 35 6 - S Vr= + o"-oo i C sin {i'V + i"(M - w") + + , ■ o" - ooi O sin (\f-- + a"), Mars. #=0 $=2D | (9= -2 D + *±z * ! * i' i" a ■ « C a c 1 a - I I S3 3° 53 7 232 / 232 3 9 5 196 1 -3 2 285 4 285 1 106 6 - 2 286-1 206 286-1 42 107-2 61 105 10 - I 72-8 271 81-4 7i 251 29 249 3 2 O 299 3 265 3 -3 3 335 n 335 3 151 5 - 2 291-6 5° 292 11 in 10 1 11 5 -4 4 194 3 194 1 5 2 ~ 3 164-8 5 1 165 11 347 15 342 3 - 2 2 97'3 82 297 18 117 9 "5 1 1 — I 3°3 1 -4 S 215 7 215 1 3i 3 -3 170 2 5 170 5 349 5 ; 349 2 -5 6 254 1 -4 42 16 42 3 223 4 | -3 176 *4 176 3 356 1 35° 2 -5 7 93 5 93 1 i 270 2 1 -4 48 '3 48 3 226 2 228 1 -5 8 280 6 1 IOI 1 ! -6 9 332 3 6 = 1 15° + -1 1 = -I 4- ty + l i' i" a c C ce c — I I 52 3 -3 2 285 2 — 2 287 8 29 104 5 — 1 67 2 8 69 3 11 — 3 3 33 2 2 - 2 292 1 4 -3 4 166 2 7 340 1 - 2 293 2 296 1 4 -4 s 213 2 -3 170 2 -4 6 43 2 Motion of the Moon, S V= + o"-ooi C sin {i'V + i"(M - m ") + 8 + a°\ =+ o"-ooi C sin ty + a"). 6 = 2D-Z 61 = = -2D + I * i^ + 2 * 4,-1 i' t a C C a c C — 2 i S3 1 - I S 2 * 12 5 2 5 O is 2 "3 2 286 2 10 - 2 287-0 2 3 117 285 3 15 — I 70-4 10 5 2 68-8 11 57 O 86 5 -3 3 33i 2 10 - 2 291 3 18 291 1 8 -4 4 186 4 -3 166 5 27 164 4 - 2 297 3 15 2 95 4 20 - I 299 2 -4 5 212 1 6 -3 169 2 9 169 4 -5 6 74 2 -4 43 2 8 42 1 -3 i7S 2 172 4 -s 7 90 3 -4 47 4 48 2 -5 8 282 3 -6 9 3 2 9 2 (5 = = 2D + Z 0-= -2D-Z <9= 4 D-/ e= -4D + 1 + + -Z * if- + Z <|< + z ,;,- if t" a c c c c C c - I I 232 I _ , 2 105 I — 2 106 2 II 2 2 - I i°5 1 5 1 5 1 I — 2 3 in 2 4 347 3 ~ 2 117 2 2 III «§PP W^W:8'' 8||t; : sp::'' ^K^-^-Bt*' Hly ^j'- - y •* : : flSffpS^ SK£ : w: : : : Sx : : M& : : : :* fg : : IS ; I'll bwb? r ^fH Biii BESjfey : :%s : : : : : gl« S» : i 1 6 4 Mr. Ernest W. Brown, Theory of the §V, SU= + o"-ooiCsin {i'l' + i"(M-za") + 6 + a°}= +o' , -ooiCsin (j + a°). i' i" = a 2^-2D c = a - 21+ c 2D c - I I 52 3 5 2 6 o 14 1 -3 2 287 2 286 2 - 2 286 12 286-9 3 42 - I 6 9 1 32 70 2 29 90 2 -3 3 332 3 - 2 291 6 291 9 -4 4 197 3 186 1 -3 165 3 166 10 - 2 295 11 296 9 -4 5 212 2 -3 169 3 169 4 -5 6 74 - 2 5 -4 42 1 43 3 -3 173 2 174 1 -5 7 91 . ■ -4 48 2 47 2 -6 8 *35 — 1 3 -5 281 1 0=2F- 2D e= - 2F + 2D e= 2 i *-F fr + F *-i i' i" a c a c a C — 2 2 286 3 286 6 I06 2 - 1 67 4 70 4 -3 4 166 1 - 2 2 95 1 2 97 1 Motion of the Moon. 65 jD/' sv= + o""ooi C sin - f i'V + t"(l - w") + + a } = + o"-ooi C sin (^ + «°), Jupiter. 6 = $±i a C a 77 17 82 9°'4 724 9C4 348-4 262 3 I2-6 2 8 359 186 . 185 1827 208 182-5 161 '6 r 73 163-0 358 14 348 102 9 105 187-0 43 186 173 23 174 101 3 191 7 191 '77 3 45 39 6 2 9 5 6 = 2 I) a C 265 t-9 269*7 211 182 39 5 10 2 *2 104 339'9 44 183 2 290 7 7 21 351 287 It 271 190-5 182 35 2 2 1) 34 46 5 2 9 6 = 1 = - - / i" a c (7 a c 2 1 83 2 8 I 90-0 27 102 2 72 14 17 4 13 8 5 16 I 2 3 2 185 1 4 2 182-4 14 56 3 2 14 1: 162 6 2 3 344 3 3 3 107 1 4 281 1 2 187 3 1 1 7 3 1 173 3 3 4 104 1 2 191 1 Royal Astron, Soc, Vol. LIX. ■■ 66 Mr. Ernest W. Brown, Theory of the S V= + o"-ooi C sin { i'V + i"( : ) - rs") + + a' / ] = + o"-oo i C sin (\|/ + a). 0=2D-Z 61 = -2D + / * ■+ + « * •J--J i" a c C a c C 2 I 84 7 37 I 89-8 82 417 90-9 10 58 4'5 ' 14 75 IO - I 16 87 3 2 185 4 19 2 182*3 44 214 3' 2 1 6 I 1 60 - 3 17 89 160 3 IS O 8 3 44 3 3 3 109 3 IS 276 1 2 187-2 9 42 6 1 I 171 2 11 179 2 4 4 18 2 3 106 4 2 192 1 6 I 177 1 - 2 - I O ^3 - 2 _ j ~3 - 2 - 1 265 270 181 S 2 34° 291 7 35* 2D + ^ * *-l C 4 7 39 1 7 2 4 19 1 8 1 4 )= - 2 D-Z 272 191 340 * *+<: c c I 7 2 9 <? = 4 D- -/ •J- <(/ + <: 6 = ^ 4 D + ^ \jl-l i" a C e a c c - 2 I 267 1 - 1 269 3 9 272 I l8l 2 J 93 2 - 2 2 2 1 4 - 1 340 2 Motion of the Moon. 6 7 :F= + o"'ooi C sin {i'V + i"{3 - vs") + 9 + a°\= +o /, -ooi C sin (^ + a). e=2i~ 2 1) 6 = - zl+ 2 I) * ty-i * t|< + £ i' i" a c c a <7 C - 2 I 72 9 84 8 - I 90-6 47 89-9 8 iS° O 8-9 2 46 6-2 2 44 I 1 3 ^3 2 187 1 185 3 - 2 182-8 53 338 i82'4 3 53 - I 1S8 12 161 2 33 O 33 1 iS 1 -3 3 100 2 108 3 - 2 S'9 94 130 187 11 - J 170 2 172 4 - 2 4 11 1 4 192 2 6=2l •J- 4--Z % i" a C a - 1 1 270 6 187 1 - 2 2 3 3 - 1 34i 1 6= -2? 4- i|< + 2 C C e = = 2F- 2D 4--F = -2F + 2D >J< + F i' i" a c C a c a - 2 1 84 1 _ 1 91 1 10 90 2 19 11 6 2 6 - 2 2 182 7 1 161 2 l60 4 - 2 3 187 1 68 Air, Ernest W. Brown, Theory of the 35 8. SV= +o"-ooi(7 sin {i'l' + i"(S-ia") + e + a c Saturn. = +o"-ooi(7siu (4r + a), + f±/ G = 2B (9 = - 2 1) ' i" a a C a c c I I 169-4 42 169 8 35° 12 2 o 3S 6 -s 45 354 16 180 5 5 2 2 339 8 339 1 '59 4 I 337 n 33? 3 156 3 117 3 2 3 340 2 I 341 2 = 1 e= -1 0-2D-/ 6= -2D i i-l 4* + 1 i, f+l # i" a a c c G C a (7 2 1 169 2 1 169 6 - I 5 24 169 O 2 2 2 10 8 2 2 2 339 2 2 9 1 337 2 1 6 2 3 34° 2 6>= 2 D + I (9= -2D + Z 6i= 2 Z- 2D 0= - 2I + 2D *-z *-l + -* * + ? i' i" a c c a C C - 1 I 349 2 169 2 II I 80 I I 6 6 .... 2 2 341 5 2 — 1 337 2 - 2 3 334 1 *-J 359, «f F= + o"-oo6 sin {V + 347 ), a« planets. Motion of the Moon. Section (viii). Final Results for the Indirect Action. 69 :6o >. Tests.— No general method of testing the work appeared to be available, but various peculiarities of the solution very much aided in the avoidance of the kind of error winch is most likely to occur — one running through a whole series of terms. In order to try and abbreviate the work, the two terms with the same argument of the form a sin \f^ + & cos ^ in the disturbing function were combined into a sino-le argument of the form A cos (^ + «) as early as possible. This might have become very disadvantageous in the final process, where terms with different values of A, « had to be added together. It was, however, obvious that for nearly all terms with the same ^, the angle a should be nearly the same ; and the cases where this was not to be expected were evident. This approximate equality of the angles therefore served as a test. Again, let us consider the terms with arguments independent of the lunar angles. They are formed of the primaries with arguments <p and the secondaries with arguments (</> + 1) - l,(<t> - 1) + 1, (<p + 2D - 1) - 2D + 1, etc. When the final addition was made, it was found that the sum of the secondaries was always small compared with the primary, unless the primary was a term of very long period— an exception of rare occurrence and easily noticed. Consideration of the peculiarities of the method of variation of the constants showed that this must necessarily be the case.* This fact furnished a full test of the principal terms whose primaries have the arguments (p±l, since these secondaries are the largest terms arising from those arguments ; it was also a partial test of the terms with arguments $±(2!) -I) in the disturbing function. The final terms containing cp + 2D arise chiefly from the primaries with arguments <p + 2 1) and the secondaries with arguments (<£ + 2D - 1) + 1. In general, the latter are very nearly half the former ; this can also be shown to be a consequence of the theory. This tests the terms with arguments 2D, 2D - 1 in the disturbing function. The only important terms not tested by these methods are those with arguments </>±(2Z-2D), all of which have periods much longer than the month. The resulting terms in the longitude which have the largest coefficients are those with arguments 0±(/-2D). L'He existence of these tests raises a doubt as to whether the variation of arbitrary constants is the best method for treating the numerous short-period terms. Possibly a direct method might be more simple. It would certainly have the advantage of finding the changes in the coordinates directly, and of avoiding the formation of derivatives with respect to n. It might be advisable to use a direct method for most of the terms, and to use the variation of the elements only for those of long period. 361. Arrangement.— The various terms with the same period which arise from f ie substitution of the elements in the Moon's coordinates have been collected into one term ; in some cases there were as many as eight such terms. * See Monthly Notices, vol. lxviii. p. 166. M jo Mr. Ernest W. Brown, Theory of the As with the direct inequalities, the terms are divided into classes, those added to the coordinates and those added to the elements. The selection is somewhat arbitrary, and it largely depends on convenience for the formation of the tables of the Moon's motion which will be a natural sequence of this work. There are no terms which seem to require the other class used with the direct terms — those partly added to the co- ordinates and partly to the elements. In other respects the arrangement is quite the same. The angles in the preceding section were the mean anomalies of the planets ; they are changed here to mean longitudes to facilitate addition with the direct terms. The terms given are those equal to or greater than o"-ooi, with the exception of a few (e.g. in the latitude due to Mars) for which there are no portions due to other causes. But all terms greater than o //- oo2 are included, except one or two of very lone- period, noted above. In the terms added to the elements, t e represents the number of centuries reckoned from 1850T). The two terms added to n't + e are to be included in the arguments D = nt + e- n't — e, and l f = n't + e' — vs'; and the secular part of w' is to be included in the latter argument. All terms given at the end of Chap. IX. which contain the arguments dtil' (1 positive) are to receive the factor 1 +i'Se'/e' = 1 — •00248*7,,. I 2 3 4 s 6 - 2 - I I 2 3 4 S i 2 3 4 5 3 5 6 Motion of the Moon. 562. <5F= +o // -ooi(7sin {9+fF + i(T -Y) + a c '}, Venus. O'O 179-8 179-6 180 259 77 323 273'3 271-8 90 93 216 180 20I'I 20 3 *9 92 115 120 )=2D c 344 5°7 5° 14 S 1 14 [ 55 14 3 1 5 129 18 12 2 19 2 4 180 2 3 2 2 10 1 180 38 1 O'O 84 2 179-6 144 3 178 17 4 180 8 S 5 6 3 7 2 8 2 232 3 2 271 40 (9= 2D ^ a 1 3 271 4 89 5 103 6 92 1 -3 - 2 269 — 1 254 1 101 2 28! 3 2 3 199 4 202 5 20 2 -4 338 -3 342 - 2 7 3 S 116 3 ^5 64 ~3 91 -4 ~3 - 2 - 1 1 2 4 - 2 - 1 1 2 3 4 -4 -3 c i i 37 - 1 - 1 S 1 2 2 1 2 3 6 4 23 5 5 - 2 -5 8 -4 13 -3 3 3 5 11 — 3 ^5 3 4 1 3 2 2 1 u u 8 90 i8o'o 61 O'O 61 i8o'o 112 4 258 4 79 1 27 3 271 33 272 34 9i 3 89 3 268 27 269 33 -5 ^4 -3 - 2 - 1 1 2 3 4 5 6 - 2 _ ] I 2 3 4 -5 -4 3 - 2 - 1 1 151 102 282 209 204 19 161 336 33 1 115 65 )=2T>-1 C 3 3 7 20 3 3 3 3 21 3 3 O'O 1 4 58 i8o'o IS 87 O'O 120 179-6 167 178 21 180 8 3 1 259 3 252 4 37 1 271-4 65 271-9 49 90 5 90 1 270 3 269 10 268-1 43 2 5i S 101 9 281 13 ■UK Up E 3gSm£>: $§:-:-:*!i : 8h«P&"'- Mii llPi * II ?:£&: •:-: ; : : :"> Bl| iBSI (£>:$ w Mi Kiwim W. |;i:r\\ n /'/./.„_,/ ,,/' '-' - r 3 4 5 - 6 :-'4- 3 _■ 5 " 5 — 3 )■-■-. -A > 200 202 •9 34-- 3.^ 34° 6 i 14 66 91 i=2D + / .,:>"•, >-.,i( 'sin ; f» +./T + .'iT- \'i + . ( ;. !'.-,,».<. .», 4 4 •i 2 24 3 4 2 j I 180 4 1 : ' : ;'i//.V'6 ;:V-' 10 -• ]So '5 3 3 4 1 2 271 5 t 3 - 1. 2 1 2 2 3 ~ 2 -3 271. 269 IOI 28l .' \ 199 v -•: 341 ; (J-4D-/ 1 = 2/. 2 -73 3 3 272 2 ~~ 3 268 2 -2 ..' ; 207 3 3 .::.: 1 -3 -3* 1 1 180 6 (9 = ^ 2/ - 2D 2 3 92 e_=2i 7 1 2 — I i _» 3 180 7 5 '\- 2 8 3 2 7 1 2 2 7 2 2 - 1 180 5 — 1 "3 211S 3 1 5 - 2 3 2 180 8 2 6 200 3 <5F= +o //, ooi(7siii(0 + 0). - 4 T- 3V + 272°'I <^ = 5T-3V + 2i6° 8 .: • /' '»..'/• • " ' ; - ' " /- l> 51 .. ' '«3 ' ^ X 9 -I) 3 2 w a ±l 3 2/- I) 2 w 8 + 2F 4 1 ' - 1 s 2° 7 -w g -2F 2 -/-- J) 3 563. SV= + o"-ooiG sin { 6 +/'M + i(M - T) + «°} , Mars. & = o /' i a c j" i a "',-:"' X^. /' t a C 1. 180 30 1 36 s 2 183 2 .' 1 So - 2 203 .1 2 2 ■;•■•; 269 2 *45*° Si 3 35'> 1 1 2 2125 48 3 25 4 34-' 3 3 5 2 4 244 16 [ ■3 2 5o 6 4 320 9 5 62 5 Motion of the Moon. 73 SV=+ o"-ooi C sin { 6 +j"M + t(M - T) + «°} , Mars. i a C /' f a c I iS° i -3 -3 264 2 3 277 14 - 1 34 2 4 276 14 5 6 275 94 6 3 6 = 1 0= 2 D - 2 5 - I 2 I 180 4 2 181 44 3 5 4 328 2 1 224 23 2 212 6 3 214 8 4 37 3 5 16 1 -3 327 1 - 2 328 3 - 1 3i7 23 139 1 2 280 1 3 3 2 244 5 3 4 4 246 4 5 59 i -4 296 1 -3 2 - 2 297 8 40 2 3 3 275 4 276 5 277 6 90 -3-4 264 Royal Astron. 1 2 2 1 1 Soc. 0=2D -3 180 3 - 2 38 — I 6 I 180 7 2 43 3 3 -3 261 1 1 224'2 66 2 212 11 3 213 13 4 30 1 S I96 1 -4 ISO 1 -3 33° 9 - 2 327 9 - 1 304-2 67 3 279 1 2 246 16 3 245 5 4 244 3 5 1 -5 296 1 -4 3 -3 29S S - 2 294 17 3 277 3 4 276 3 -4 264 3 -3 263 2 o -4 - 2 , Vol. LIX. 2D -2 19S o 3 11 > = 2 D-Z i a 1 O 3 1 180 8 2 181-0 61 3 353 5 4 329 1 3 259 2 1 220 31 2 212 11 3 214 14 4 27 3 5 164 5 3 327 2 2 328 6 1 320 35 139 1 3 280 4 2 244 11 3 6 4 245 5 5 60 1 6 3°9 3 4 296 1 3 3 2 297 14 3 275 1 4 2 5 276 1 4 264 2 3 266 2 > = 2 D4-Z 180 o , 82 212 214 93 6 I 3 1 2 3 10 mini 74 Mr. Ki:n i>t \\". I!i:.)\v\ : '/'/„;. ,-,/ -,/ ,'/,. SV= + o"-ooi C sin {6 +/'H + i(M - T) + «°}, Mars. !=?D + J ?=2? ^2?-2D «" o ' ' ■ i" W0^~iM: ;;i:^ : ::^y::^:\y::i c •/'• •t a C 2 2 245 . 1 - 2 9 '. 3 — 2 O 4 2 — 2 297 1 ■ 2 ISO 3 1 1 87 1 1 I 232 3 5 196 .: <9 = 4D-£ ■/■/■^^■::}:y; : ::^ - 1 - I 308 3 - 1 - 1 98 .■; O 2 2 2 6 51 1 364. S V= + o"-ooi Csm{8 +fj + i(,J - T) + «°} , Jupiter. 2D 6 = 1 i: ;vK;|;s:«j§sj!;|| C I 178-9 , 712 2 '359*6 200 3 7 10 -3 .. 2 57 : 6 -2 '274 18 33< v 3 259 1 238-0 170 2 35 2 44 3 355 4 - 1 25° 10 334 14 1 238 2 5 2 344 6 1 230 3 ) = 2D ■3 : I - 2 180 25 - I 1 31 I i 7 s- 5 J.67 2 359'2 87 3 r 3 . 7 4 . 12 2 3 19 29 1 237 35 2 352 *5 3 358 2 i a - 2 . 9 10 - 1 303 6 O : ' 184 . 35 '2 '• 273 9 3 102 .6 . . 35i 'i'' 1 236 5 2 345 3 - i 288 1 '• 200 . 3 1 no 6 1 230 1 )=l -3 173 • ■-% - 2 180 40 - 1 i*o 136 1 179-0 i5° 2 ' ' • 180 19 3 .25 3 - 2 274 6 * 0' : ." 298-5 71 1 239 40 2 35 1 38 3 • 257 i - 2 188 7 - 1 301 36 3" * :;:«;:; C - 1 . • 24i's :- 2 273 6 ' -2 ' — I 250 2 326 7 I 238 5 .".2 344 2 - 2 — 2 . 196 .! — r 302 5 214 7 1 • 290 2 = 2D-/ -3 *75 2 — 2 .. . 180-4 333 - I 1 ■4-4- I 178-4 2 11 2 359'2 s 9 3 14 6 -3 z6 3 1 ;_ :2' ; 279 9 • :-P • S'S' 56 I 237-0 46 2 352 20 -2 ' . 9-1 130 _ I" '. ' 301 '3 O : 174-2 60 2 273 16 Motion of the Moon. 75 1 j:^:lljjl|j:^^;:;:IE::; ■■ S V= + o"-oo i C sin {0 +/'J + i(J - T) + a°} , Jupiter. »:| 1 6=2l)-L e = 2l) + i 8=2l-2~D j" i, a j" i a. j" i a 1 ~ : HHilll1IL-::: : : : fri>>; : <??: : S - I 3 102 7 -12 273 3 0-2 180 5 1 y:H»ffl££:3-{g:2:-:*: 1 2 o 35i i 3 102 I - I 2 11 MB ^SUf; &' 1 I i 237 6 I O 3 S - 2 2 344 3 -2 196 4 1 358 9 2 1797 i 2 I 7°"9 53 I 94 » - 1 305 2 2 179 5 - 1 - 1 302 2 HI IBKllill 171 1 1 172 1 186 2 fig |:aHH& 1 r § -JH 1 291 3 1 57 1 — 10 2 1 2 2 164 6>=2Z- 4 r) I ■ 1 Hk ' ! Jh e=2i) + i (9=2/ 1 2 171 1 Mi :.-MKflBre : >: £$:•: ]3»fl O -1 1 5 O -2 180 3 1 178 21 - I 2 II <9=2F- 2 D ^Ssliift:^ l« 3S 2 359 8 I 178 12 - 1 2 H 3 r 4 1 203 1 180 1 ■ I 353 4 1 237 s '10 293 5 I 239 3 ffii : « 2 35 2 3 2 171 I # = « 3 m^^^^K^^-'-^»^bB - I - 1 3°3 1 -I -I 301 3 10 3 1 Sra«3»§<^i»- : -:-i ^9aH 182 4 O 247 5 2 168 6 I «HBBWBk>:?¥x-:': 3K&H 365. cfF= +o"-ooiCsin {6+j"S + i(S- T) + a°[ , Saturn. flippy 19 6 = <9=2D e=2D-i ililpspi IH i" i a C j" i a C j" i <* c H BSlSSKg: * #£|:o 3rH: o i 179 6 42 1 I 257 3 O -2 l8o 5 H 208 2 270 2 - I 2 '((■■f If iff- 1 ■ iS I 266'6 45 -IO 255 4 I 180 14 H IMH9 1 257 13 2 280 2 2 O 4 H: iltl 'it 2 270 2 I O 4271 6 Hi j|J|t < £ ; IB 2 297 3 6 = 1 2 257 3 ■si H&Kffifi:! S&iHf i 171 2 -2 180 I - I - 2 277 1 ■■11 jl||p|||l ||H| - 1 6 O 267 6 Hi Bjragroj^K&xixStwSE: = 2D 1 180 10 I - 1 6 1 1 180 10 205 270 4 1 265 15 1 257 2 -1 -1 283 3 275 15 6 = 2~D + l 1 180 2 :Bk| [it < 1 m jl ailifififll |H9 7 6 Mr. Ernest W. Brown, Theory of the 366. All 'planets. S V= - o // -02 4 sin (V - 5°) - o // -oo 4 sin (l'±l - 5°) - o"-oo2 sin (2' + 1 - 2D - 5 ). Latitude. 367. <$£/= +o"-ooiCsin{0+ i ; , T + i(T-V) + a o } = + o"-ooi Csin ^, Fcmwj. = ±F J ?. a I O 2 180 I 2 2 73 3 272 2 3 201 ?= -F + 2D a 4 7 6 2 2 2 7 I 180 M I 16 2 180 23 3 2 2 271 9 3 272 6 3 268 2 2 269 6 1 101 1 2 280 1 3 199 3 3 34i 5 (9 = = F + 2D i '' a - 1 180 1 2 180 3 1 2 271 3 272 1 - 2 268 1 IOI 2 280 2 3 199 2 ™3 341 6 = ± F + / c* 2 5 9 1 2 2 1 1 1 1 2 I 2 2 180 4 2 271 1 3 272 1 3 199 1 _-_ - F + 2D + 1 f9 = F + 2 D + Z i i E I 2 180 (9= ±F + 2 D-J o - 1 1 2 2 3 - 2 180 271 269 199 ? M-5 T -3 v +2i5°-6 'A ~5 T + 3^ + 337' "'1-2D + 5T-3V+36' u\ + l+ 5T-3Y + 216 3 «W+5T-~3Y + 3 6' Wi + ST-sV+^s" ^-81+ 5V + 67° 3 3 6 8 3 2 1 c 77 3° 3 4 4 3 7 368. r)Z7= + o"-ooi Csin {d +f'M + i(M - T) + a }, Jfars. ^=±F 6 = F- 2 D J' I a a 3" i a 2 180 3 - 2 2 ISO = +F + /-2.D 1 1 223 ~ 2 % - 1 - I 316 2 2 269 Motion of the Moon. 77 369. SU= + o"-ooiC sm{6 +f'J + i(J -T) + a } = +o // -ooiCsin ^, Jupiter. 6 = + F = = F+ 2 D 0= +F + Z-2D j" i a c /' i a j" i a a I 180 9 I 180 8 - 2 l80 4 I O 48 8 2 3 - 1 O 11 1 O 350 1 1 180 2 6 = -F + 2D I 237 2 2 O 15 2 352 2 1 357 2 - 2 — 1 O 3 6 - 1 O 2 2 171 6 2 273 2 - i - 1 302 2 1 180 29 3 102 1 186 2 2 O 12 I 35° 7 0= ±F + 2I — 2D 1 237 6 = --±¥ + 1 2 357 2 2 3S 2 2 - 2 2 1 2 354 4 I - 1 3°3 2 - 1 6 181 4 1 180 8 * C 2 273 3 2 1 Wj, + J 5 3 102 1 1 301 4 ?,«! + 2J+ 168° 35 1 239 2 W t - 2J + 24° 18 = F4-2D 2 35° 1 «l + 3J + 156 2 - 2 4 - 1 _ j 301 2 «'l + I+2J + 168 2 - 1 180 3 240 4 »1 -Z+2J + 348 2 370. Terms acWecZ to <Ae Elements. Sw>!= +o" , ooiCsmi/f+s"'82!f 1 , 2 i 3 T-8V+ 3 i3°-8 Z+i6T-i8V + 33i° Q-4T + 239 8M-4T + 310 9 M- 5 T + 3os° 10M-6T + 306 HM-6T + 33S 13M- 7T+ 19 1SM-8T + 43 17M-9T + 63 W s +2 76° - 2 C 234 6* 3 3 8 2 6 6 26 4 289 8 ( w i - "'2) = + i6"4 + 44"'i4 2 + i"T29 sin («« 3 +276°-2) 8w s = +s\ + 6"-46!! c 2 + is"'59 sin («%+ 27 6 °' 2 ) Sy = +o" - 698 cos (» 3 + 96°-2) S(w'tf + «')=+ !"" 8 9 sin ( r 3T - 8V + 134°) + o"-20 sin (isM - 8T + 216°) 8w'= 4-0° -32^ 8e'= - 'oo248e'4 Mr. Ernest W. Brown, Theory of the CHAPTEB XIII. ACTION OF THE FIGURES OF THE EARTH AND MOON. Section (i). The Disturbing Function for the Figure of the Earth. 371. Let A, B, C, I be the moments of inertia of the Earth's mass about its three principal axes at the centre of mass and about the line joining its centre of mass with that of the Moon ; E, M the masses of the Earth and Moon. Then it is well known that the disturbing function R is given, with sufficient approximation, by R=(E+M) A+ J^ + S ) S3 I , , . . . . (1), 2r s E since the next term of R will have an additional factor of the order 6cr 2 , this being the approximate ratio of the square of the radius of the Earth to the distance of the Moon. It is true that there is a term with a factor of the order 60 ~ 1 , but this term is exceed- ingly minute, owing to the approximate symmetry of the Earth about its principal axes. Let V, U, a, S be the longitude on the ecliptic, the latitude, the right ascension reckoned from the A-axis, and the declination of the Moon; -^ the precession, e x the obliquity of the ecliptic. If P be the pole of the ecliptic, Q that of the Earth's equator, the parts of the spherical triangle PQM are : PE= h , QM= 90 - 8 , MP= 90 - U, QPM = go° -V-if,, and therefore Also so that The second term of this is quite negligible : its principal arguments have daily mean motions of the order 3. io 6 seconds, and A - B is known to be very small compared with C - |(A + B). Hence R = (E+M)'?£(§-sm*8), where «V = JL(o - ^±^"\ . '. . . (4). sin 8 = cos €j sin U+aine 1 eos £7 sin (V+ij/) ..... (2). I = A cos 2 a cos 2 8 + B sin 2 a cos 2 8 + C sin 8 8 , A + B + C-~ 3 I= 3 (c-A±I J Vi-sm 2 8)-f(A-B)cos2acos 2 S. . . . {3). Motion of the Moon. 79 372. Transformation. — -Since p cos ( V - T) + i,p sin ( V - T) = u, s , r sin U= z , r cos U = p , equation (2) gives r sin 3 = z cos e,+ — sin e 1 (we( T +W t - se -(T+W 1 ) , 21 1,2 _ f 2 s iu2 S= (|)' ! - z 2 )(i - § sin 2 fl ) + J sin 2 fl (aW+«' + s%- V+»') - 1 sin 2<r 1 (we<T+<M' - ge -(T+«t). r Fhe last two terms are the real parts of \ sin^ . w 2 e 2(T+ ' w ' , zi . sin 26! . we (T+,w \ Hence ii? is equal to the real part of where :*{ m-^t ■} ^.(T)'M + ^r#-« Mr 4 /* / -s- sin z «, u,= -i~sm z e, , ij., = l j sm2e 1 6 vi 2 in 1 fi being given in (4). The values of the Moon functions have been given in Sect, (v), Chap. X. All the terms have the factor p., which is treated as a small quantity of the first order, and we should properly put e x = const, and ^ = o. But it is convenient to retain the mean motion of ^, as this motion affects the arguments to a slight degree, and is retained with- out any increase of labour. The m» are then constants which take the place of m"/m! in the equations of variations. In forming j u it is to be remembered that m contains n 2 . Section (ii). Numerical Results. 373. Adopted values of the constants. — I take giving £] = 23° 27' 32", daily motion of \\r = +o" - 14, /x=+[7'6658] M^+tS'925 1 ]. ft=+[4'3 8 36]. ft=+[S'4i9i] .... (8). The only one of these constants of which the value is doubtful, within the limits of accuracy required here, is fi. It will ultimately be determined by the coefficient of the argument w x + ^ in latitude, the principal term arising from the figure of the Earth. 1 adopt a value here to correspond with that marked (/?) in my paper " On the Degree of Accuracy, etc." * ; this is obtained by comparing Hansen's observational value with Hill's theoretical values (which closely agree with those obtained by me) for the coefficient in question. 374. Final results. — I omit terms whose coefficients are less than o ,/- oc>3, and obtain, for the terms in longitude, &V= +o" , 020sin(2D-£) + o"*oo4sin (2F-Z) -o" - 038 sin (2W 1 + 2f- 2F); ; mm * Monthly Notices, vol. Ixiv. p. 531 8o Mr, Ernest \Y. Brown, Theory of the in latitude, 8U= + o""o83 sin (210 x + 2f - F) - o"'oc>3 sin (210 x 4- 2f - F - 2D) ±o"'oo5 sin (2W! + 2i/f-F+Z)- o"-oi7 sin (wj + i/r) - o" , oo7 sin (wj + i/r - 2D); and, added to the elements, Su>j = + y"-^ 1 7 siu (w g 4- f) , 8w 2 = + 641% - 2"'092 sin (to 8 + f) , Si« 8 = - 600'% + 96"'69 sin (10 3 + f) , S» = - o" - oo9 cos (iv s + '/'), Se = + o"'oo2 cos (a> 8 + \j/), Sy = - 4""3 5 1 cos (ie 3 + xj/) , of which <$», Se may be neglected. The principal term in latitude which results from these values is — 8 / '*355 sin (w 1 + ^ ). Section (iii). The Action of the Figure of the Moon. 375. The Disturbing Function is of the same form as* that for the figure of the Earth. Let a' denote the longitude from the A'-axis on the Moon's equator of the projection of r on this plane, and S f the inclination of r to the same plane. Then if A', B', C ; , V be the moments of inertia, the C'-axis being that perpendicular to the Moon's equator, and 1' the moment of inertia about r, we have, as in § 371, A' + B' A' + B' + C'-3l' = 3 C (C' - ^±Z)(! - sin 2 8') + |(B' - A') cos 2a' cos 2 8' . (9). 376. Transformation. — Now the Moon always turns the same face to the Earth, and, if we neglect the small real (not apparent) librations, its angular velocity about the C ; -axis is therefore constant and equal to n. Moreover, it is well known that its equator and the ecliptic intersect in a line whose longitude is w s ; call this point on the celestial sphere Q. The mean angular distance of the A'- axis from Q is therefore w 1 — vj s . Hence, from the right-angled spherical triangles having each a side, one on the ecliptic and the other on the equator, and a common hypotenuse QM, cos 8' cos (a +w l - u> 3 ) = cos U cos (V - w a ) . If we neglect S\ IT, this gives </= V—w x . Put a' — V—w x + oV; then oV depends on squares of S', U, and cos 2d = cos 2( V— w x ) ~ 2S0.' sin ( V— iv t ). As we shall neglect quantities of an order higher than the second with respect to the eccentricities and inclination, and also the inclination multiplied by mr, we can neglect the second term of this last expression. Also if 7 = sin ■§•»', and if —i x be the inclination of the lunar equator to the ecliptic (it being well known that the ecliptic lies between the, mean lunar orbit and the lunar equator), we have with sufficient accuracy siu 8' = sin (i + ?',) sin ( V. - w g ) = sin (i + i t ) sin (w x - w a ) . Substituting, we have, for the disturbing function i - sin 2 (i + n'l) sin 2 (v^ - w s ) + (/*"///) cos 2 ( V - w,) { 1 - sin 2 (i + %) sin 2 (w t ~w a )} R = (E+M) 0?jX where 2M A' + B' a 2 fi' = -3_(B'-A'). Motion of the Moon. 81 ^•jj. Form for computation. — The principal periodic terms have all short periods, and we need only consider the constant parts which give small constant additions to b.- n b s . Now, if we retain quantities of the orders previously noted, that is, those parts which are of the second order with respect to e, y, the portions depending on e 2 will alone affect h, and those depending on y 1 will alone affect b 3 . Hence for the former we can put and for the latter, R = (E+M)^[y + n" cos 2(F-w,)]; R = (E + M) a f - \ix - !//'] sin 2 (/ + «',) , in which terms of the order fm 1 have been neglected. Let p e be the coefficient of e l in a s /p 3 , p c that in a?p~ s cos 2 (V— iv-f or in a s p~h(^' 2 . Then, referring to the disturbing function for the figure of the Earth, and remembering the formulae for obtaining Sb.,, $b 3 (§ 270, Chap. X.), we see that the values of §b 2 , Sb s for the figures of the Moon and Earth are respectively in the ratios 1>-Pe + 3p"p< '■ K J ~ f s i u ' 2 6 i)/Y nn d (l J - + p')-y. s ' n2 ('• + b) '■ l J '( T _ f sln2 6 0^-. s " 12 i ■ 578, Numerical results. — From the results in Sect, (v), Chap. X., and in the first section of this chapter we have A- + '386 , flf =--6j8, /,,= +[7-6658], 1 -#sin 2 £ 1 = +7623, S6 2 =+6"-4i, 8l> 3 = - 6"-oo , the last two being the values for the figure of the Earth. Hence, for the figure of the Moon, the annual mean motions in seconds of arc are S&.,= 8"-4i^-44""3— , 8/ ^ ; t x /J. 7 ".g„/* +M sin 2(i + h. fl. Sill 21 f shall now assume that the Earth and Moon are of similar constitution, so that C fl : C;E in that ratio of the squares of their diameters, that is, as ('2 jff : 1. I also take A + n lence 2O „/ A' + B" T Q E I 1 - — = •00328, *'= 5 c "i , ?'i=i L 'S (§379) • ,/ir - a'\ A' 37c), Adopted values for the mechanical ellipticities. — -The results of Dr. E. Hays,* for the lunar librations, give i, = i'-32', B'- A'-= +-OOOTS7CA C- A' =+ -0006290' . If we accept these values, we obtain for the annual mean motions 0/)., -- + o"'o3 , So., = — o' ■ 14 . In order to obtain these quantities accurately to o"-oi, it is necessary to know the two mechanical ellipticities within 5 per cent, of their true values. * Abh. der Math.-Phys. El. der K. -Sachs. Gess. der Wiss., vol. xxx. (1907) p. 69. HoYAL ASTROX. Soc, Vol. LIX. H Nil WA 82 Mr. Ernest W, Brown, Theory of the CHAPTER XIV. THE REMAINING PERTURBATIONS. Section (i). Corrections due to the Masses of the Earth and Moor /2~/3 380, Correction due to the substitution of m' instead of m' + JE + M for n n a This is noted in § 4 (a), Chap. I. It amounts to diminishing the disturbing function due to the Sun by the factor i-(E + M)/m f . The method of Sect, (iii), Chap. XII., might be used, but it turns out to be troublesome. It is more simple to use the ordinary method for the indirect inequalities by putting Sp' = [E + M)/^m\ SV = o for periodic terms, and R ~ - (E + M)F Q jm' for the constant term, where jP is the constant part of the disturbing function due to the Sun. 381, Results. — I find, as in an earlier paper,* for the annual mean motions, 8& 2 = - o"*69 , S/> 3 = + o"'i9 , the constant changes of n, e, 7 being insensible in the coefficients of the periodic terms. The periodic changes are : 8V= - o" , oo7 sin 2D - o"'O20 sin (2I) - I) + o"'oo3 sin V + o"'ooi sin (1 + I' — 2D) . 382, Corrections noted in § 2, § 4 (c) of Chap, I. — The former gives §b z = — o" - oi, and the latter §b 2 = +o"'02, §h s = — o" - oi. Section (ii). The Terms of the Second and Higher Orders, 383. Sources of the terms. — In the four last chapters we have computed the perturbations due to various causes, on the assumption that certain factors which multiplied the disturbing functions were so small that their squares could be neglected. It remains to examine with some care to what extent this assumption is justifiable, and to correct the expressions in the cases where it is not true. Let E(r', V, z') be the disturbing function due to solar action, R P that due to a planet, R E that due to the figures of the Earth and Moon, and R e that due to the motion of the ecliptic. Then if SV, § 2 V, h' be the terms of the second * Monthly Notices, vol. lvii. p. 567. Motion of the Moon. 8- order in the motion of the Sun, the complete disturbing function for actions from all sources, except that in the main problem, is m=R{r' + %r' + W, V + 8V+SW, z' + 8z')-R(r', V, o) + %R P + R E + R e . . . (i). We have previously neglected quantities of the order (SB)' 2 , and have used elliptic expansions for the coordinates of the Sun and planets in the last three terms. 384. It is convenient to divide the perturbations of the second order into classes according to their nature or the sources from which they arise. I denote by § a perturbation of the first order, and by P one of the second order. The several portions of S 2 R to be considered are as follows: — (A) Terms due to the substitution of c { + Sc ( , w, + hv { instead of c t , Wi in the right-hand members of the equations of variations. (B) (c; dr V oz -hV . + e .—\Ii. oV oz J (D) Additions due to periodic perturbations of the solar and planetary coordinates in z,lbp. (E) Changes in §R due to secular or quasi-secular variations of quantities which have been treated as constants. (F) Changes due to the secular variations of the solar and planetary arguments. (G) Third-order terms due to large second-order terms in the solar and planetary coordinates. (H) Secular variations of the second order in general. It is obvious, in the first place, that the only possibilities we have to consider are secular terms in tv x , w 2 , w s , and terms whose primaries are of very long period. In the second place, it is to be remembered that the variations of the elements contain terms of two kinds — those multiplied by the period of the primary, and those multiplied by the square of the period ; and that the latter, for terms of very long period, are large compared with the former. Hence, except in the cases of arguments independent of the %i\ (in which case the latter terms will be shown to disappear), it is sufficient to discuss only these latter terms. Finally, the greatest effect of these terms on the coordinates occurs through the change in w u so that it is generally sufficient to discuss t^w v The various classes are considered in the following paragraphs. 385. (A) The canonical equations of Sect, (i), Chap. X., will be used and developed 111 a general manner for the second-order terms. The chief results obtained under this heading, namely, a proof that such terms are insensible, practically consists in snowing that a given argument arises in two ways, and that, whenever the two illil i 1 8 4 Mr. Ernest W, Brown, Theory of the parts might be separately sensible, they are opposite in sign and nearly equal in magnitude. Let S^Cf, Shi'i be the additions to c i} w t due to the substitution of c t: + <!c £ , ti'i + Swf instead of c i} iv { in the right-hand members. Then the equations for the determination of these additions are >-$■ ***-- fW£ s 'A^i) !s - ■ ■ ■ » Let the first-order variations §c i} hv { be due to a term * in R A cos <f> = X cos (pi + a) = A cos (J^'Jj +i 3 «2 +j s w B + f), where 4^ is independent of the iv { . Then Sc^jf- cos <jf> , 8^=(- ^ - - 1 - — ) sin ^ . , . . . (3). Let any other term of R be A' cos <jfc' = X' cos (p7 + a') = A' cos (Ji' t i'i+h w 'A + Js' w s^ f') > where \|/ is independent of the w*. Let us consider these two terms alone ; then the terms in §%, §*w { will have the arguments <p±<p'. Put Then .3.3 . 3 _ 3 . , 3 . , 3 .■ ' 3 _ 3 Jl 3^ +)2 3^ +Ji dc~ B ~ 3c ' Jl &; +Ja 3iT 2 +Js 3^ ~ 8? s 3fi _, /3 2 5 s , 3 2 £ » °— =Si(r— „ t,C k + -—^ 3.«»i (4). .,A dA' . ,, , ,,/AA' dp A' JX\ ,, . , = ->; - . — Sill COS <£ -Ji I _ ' - — COS (£ Sill <£. p rfc \jp» dc p de/ To this must be added the term arising by making §w { , §Ci depend on (p\ and the derivatives of R on (p, that is, by changing the accents. We thus get ! 8% < A'— f i 4 - it) -j^L d S-- de \p' p »' 2 ' de sin tj> cos <f> + ' similar term with accents changed and therefore, since dbi/dc k = db k /dci, db t dt Z,£*-i ,/iX_,rfAVi d// _ 1 djA ..,/ i dp dp 1 dp' dp A dc dc/Kp ' dc H p ' dej \p 2 ' de ' rfeAp' 2 ' de ' dc t . sin (</> ± $') , Denote the right-hand member of this equation by F t sin ((f) + <p') + F- sin (<p - $'). * The symbols ji, j 2 , j s are the same as the %, i lp i 3 of Chap. X. ; the j 1; j 2 , j, i of that chapter are not needed here. Motion of the Moon. 85 Again, <3>R XSCfil ,i 2 X' X d 2 R dc, * \ocfiC-,; GCjCWj. , ,, JX'/ X fte 1 dX\ •,-,/,, • -i 4. ■ cos <f> cos rf> - — I — . — — — . -, ) sin <p sin <p + two similar terms dalCf p dCi\p 2 dc p de J _ p dcde. , X' d 2 k , f dX'/ X dp 1 <iX\ similar 11 , , , ,, - + —,. — -, — ± { — — . — - — . — , + i ^ cos (<£ ± th ■i p dc'dd \dc t \p* dc p dc'J term )J vr r which is denoted by G { cos (cp + <p') + G{ cos (cp — (p'). Finally, 'A ^Lj, d \ 2 7 XXV x-> • > d 2A XX' d 2 j/ cos <f> cos </>' = iZ^ cos (<£ + <£'^ vfcj/ pp \^™ J k dc k /\ J -— l k dcj % pp dcdcf, Substituting these results in the second three of equations (2), we obtain S 2 m> 4 = I (Hi - G{) cos (<f> + <p')dt + I 1 1\ sin (<f> + <f>')dt + two similar terms H, - G, F, sin (<f> + <(>') + H { - Gi Fi j sin (</>-<£') (5)- p+p (p + p'f) xr ' r ' ' V p - p (p- p'f Since we have only to search for terms of very long period, the most important terms are those which acquire the squares of the small divisors pzhp', and therefore the coefficients F, F' are of chief importance. 386. (a) If p, p' are independent of the w i} we have S\ = o, S 2 w t = o. 387. (/3) If ji=j! and p—p', the argument cp — cp' is of long period. In this case we have dp _ dp dp _ dp' _dp _ dp' dCf dc,/ dc do dc dc Hence F' contains the factor p —p', and c£%. only acquires the first power of the small divisor p —p'. In no case, therefore, can the small divisor appear in its second power for terms independent of the lunar angles. 388. (7) Ifji=j{ and p^ —y', the argument cp + <p' is of long period. Then the possible combinations from the first-order terms show that one of two things must happen : either p, p' are not small, in which case A, X' are so minute that there is no possibility of a sensible coefficient ; or p, p' are themselves small. In the latter case, the second term of F t is the principal one, and it gives for the coefficient, since p= —p', XX' 1 \fdp\dpj_ X dp\f X' dp'\ p 2 p 2 ' dc)\p' 2 ' dcj(p+~j7j 2 " ' (P + PV\ P 2 1>" 2 / V'' JdCi \p- This is of the order §w t . o'iv t . p 2 /(p+p'f, where Sw it <$'w { are the first-order terms with the arguments (p, cp'. If C, C are the coefficients of Sw ( , Sw/ expressed in seconds of Me, the order of the new coefficient, also expressed in seconds of arc, is - C -V-(-P-)\ 206265 v +pd Ss-i ; :ffi^ III '•:♦ wm-m 8 mm iii ■:>•;.::->■:: mm- mm ,¥z3 mm mm. mm iSSS:' : :*$ W:-fSW ■jvfS sbss; ■> mm :■■:&. wm II 11 Sssi¥: mm mm 11 mm Wi Siisi llli :->*•:■ mm •:•:■:•:■:•:•>: :•:•:■:■:■:•:■::■ -;^>'^;^;P II II :■-■.■■.•:•:■■•:■. mm ■:■'•'■'": : ; : : ' : : : ' : :": : :-:i :£:■:•!-:-:-'■:■ 11 mm lllllli-i ;ii mm SK* mm mm 'Mmmi HIIIII mmmm mmm mwmmi 86 Mr. Eenest W. Brown, Theory of the 389. (S) Ifj\=j 1 ' and p±p', the terms in F{" which have the divisor (p—p'f have also as a factor one of the derivatives of 6 2 or b 3 to c 2 or c 3 , that is, the factor -to 8 . If j9= —p' t the argument in (7) still holds for the principal parts of ^w x , the most- important variation. 390. (e) If j l = o, and p=p', then — F//(p -p'f becomes Hf = o, and X dX dp 1 . , , , / \ dp\f p P «'- *A {P-P? ' "A r dcPXp-p (J- v -p' X dVC d,X 1 cZA.'" j? " dcdct ' de,i ' p ' da _ 1 ^_ j p d I X dX'' p-p 2 p~p" de}\p'*" do, The principal term is the former of these two. Other cases are treated in like manner. The net result is that terms independent of the lunar arguments only acquire the small divisor p - p', while those containing the lunar arguments are at most of order (Sw^S'w^Kp-p' 2 ). A somewhat closer examination is only necessary when the latter class of terms cannot be neglected at this first test. No sensible terms have been found. 391. (£) For secular terms, we have 8c i: = o, dB/dw^o, and therefore § 2 c t = o, Shv-i — o. However, certain changes have been made in the arbitrary constants givino- changes o c f . But these only produce constant changes in the d^w^dt, and therefore * further changes in the arbitraries ; the latter are insensible. But secular terms may arise from the combination of two terms of the first order of the form (a + bt) sin (p, (a' + b't) sin <p', where (p, <p' are either equal or where their periods differ by so small an amount that <f> — <p' may be treated as secular, and a, a', b, V are constants. Owing to the minuteness of b, b', such terms are entirely insensible. 392. (B) These terms are treated in exactly the same way as the indirect terms of the first order. They are computed in § 402 below. There are no sensible portions depending on <V. 393- (0) These terms are of the order of the indirect disturbing function of Chap. XII. multiplied by 3$/ or SV. If p be the mean motion of a term obtained in that chapter, and p' that of a term in 3p or V, the worst case is easily seen to be of the order 3.10 p 1 p . ■■-- r - F - . — — ,= • —i — ; seconds 01 arc , 200205 p-p 7000 p-p for terms independent of the lunar angles, since the largest term in Sp' or SV is less than io", and there is no term due to indirect action having a coefficient so great as 2" . For terms dependent on the lunar angles we obtain the order ' ' seconds , 7000 \p - p and a brief examination is necessary. It is easy to see that, if w x is present, op', SV must have periods comparable with the month, and must, therefore, have coefficients tel* Motion of the- Moon. 17 too minute to produce sensible effects. When w 1 is not present, there are only one or two terms to consider (the most probable of which arise from the action of Jupiter) ; these were examined and were found to be quite insensible. 394. (D) These inequalities cause additions >mm ;'§al :' l??X| nISI III!! : -tew**:*:-:! sm-- 8p'(-3-/)+8p"-/+8(F' %R P for inner planets . (6), S 2 i? = 8p.I+8 P "(- 3 -I) + 8(V'-r") A+AY de drn'J_ %R P for outer planets , (7), where Sp", SV" refer, in any term, to the same planet as that to which the R P for that term belongs : R P is confined to that part independent of a a . and / is defined in is 2 96, Chap. XL To obtain the order of magnitude of the coefficient it will be sufficient to consider terms for which / or the derivatives with respect to e', ts' produce a factor not greater than 20. Also, since Sp', Sp" are of the same or of higher order of small quantities than SV, SV", we use the latter. Proceeding a.s in case (C), we obtain for the maximum order of Shv x SV-S(V'-V")f p \^8V-8(V'-V")( p p-pJ -P-P iiii ; : ; : : : : ; : : : : : : : : : : : ; :;' : :v : :: 111 ill III in which every term, as w r ell as the result, is expressed in seconds of arc. For inequalities independent of the Moon angles, the factor p/(p ~p') only occurs in its first power, and we use Sp' or Sp" in the place of S(V — V"), This method provides for terms which arise with a sensible first-order term, that is. with a term in SV already obtained. But there may be other combinations due to terms in S V previously neglected. Since S(V — V") must be very minute for terms of short period relative to the year, we need only consider terms of long period in SV relative to the month, and from such terms those containing multiples of u\ higher than the first can be certainly excluded. A detailed consideration (which will be omitted) of the manner in which the various quantities enter into SV gives a maximum value for Sho lt in the case of Venus, of about 2oS(V — V')j=?, where s is the number of seconds in the daily motion of the angles <p±(p', and SV, SV" express the number of seconds of arc in the coefficients of the terms in V, V" under consideration. 395. First let us exclude from SV" all perturbations but those produced by the Earth. Then the maximum value of S(V — V") is 10", and if we neglect coefficients hi S i io,_ less than o^'i, we obtain a maximum value for s of 44". Similar considerations for all the planets give numbers exhibited in the following table, which shows the maximum value of s for different combinations of SV with R P or R % . In the principal diagonal, where the suffixes are the same, S(V - V") must be understood; the suffixes denote the sources of the terms. Values of s less than 1" are put down as zeros. The last row T refers to the ellipticity terms ; R e is too small to give anything sensible. lllllli MM islili "■■■>»■» Mr. Ernest W. Brown, Theory of the Maxima of s for a Coefficient in $V of o"-i. sr; *f7 8r V « J/ S ' »'V B, 44 44 24 : 24 2 \R, 44 6o 24 ; 24 2 #M 1 7 ? 4 O A\ 14 14 7 I 7 O H q 7 7 4 4 2 II. 44 44 7 14 / Since the coefficient in <? 2 F varies inversely as the square of s, this table shows that for a coefficient of 1" in § 2 V the value of s must be less than 6", or the period greater than 600 years. The numbers in this table are only rough approximations ; but even if they ought to have been twice as great, it would simply have meant that the corresponding coefficient in S^V should be taken as o ;/ '4 instead of o"-i. It will in any case retain any sensible coefficients. In fact, for periods of over 100 years or so, the minimum sensible coefficient will certainly be less than o"-it„ where t„ is the number of centuries in the period. The various combinations have been considered in the same manner as that employed in Sect, (iii), Chap. XL There appeared to be only one which might, give a sensible coefficient, namely, the combination (/+i6T-i8V)-(r3T-8V) = / + 3 T-ioV, which has a period of 1900 years. This is therefore a second approximation to a term due to direct action and given in Chap. XL There are no sensible combinations of terms independent of the lunar angles. 396. In the above, all terms in SV", 3p" which arise from planets other than the Earth have been omitted, but for these omitted terms the mutual perturbations in the lunar disturbing function cancel one another very nearly. For example, the great in- equalities in Jupiter and Saturn with argument 58 - 2J appear in both i? s and E, h Now, Leverrier has shown * that the effect of such terms on the motion of the Earth may be neglected provided we take J, S to represent the mean longitude, as affected by the great inequality instead of the mean longitude alone. But, for such terms, the lunar disturbing function for the direct action only differs from the Earth's disturbing function by certain constant factors and by certain operators k + k 3 a'd/da' + h i (a'd/da'f, k , k t , & 2 , . . being certain constants independent of the disturbing functions (Sect. (ii). Chap. XL). The terms produced by k nearly cancel one another, while Jupiter and Saturn are so far away that the other terms are very small. A rough calculation shows that terms from this source are quite insensible. * Ann. Obs. Paris, vol. ii. HP III 1 SB mi 11 i IP Motion of the Moon. 89 Hence, the terms of this nature are sufficiently accounted for by adding the " planetary perturbations to the mean motions of the arguments of the planets. 397. (E) For these secular terms, the investigation of Sect, (iii), Chap. XII., shows that it is sufficient to insert the disturbed values in the final results. The only term which can be affected is the great inequality due to Venus. This depends chiefly on y"°- and tyj"/' = tcl ZA-OO, where t e is the number of centuries from 1850-0. The : , maximum change is therefore less than o"'oi per century and is quite inappreciable. 398. (F) These motions should properly be inserted in the arguments before ' .division by s or s 2 , and this is sufficient to account for them. There are only two terms which can be sensibly affected, those with arguments Z+16T-18V, and ■/ + 3T-10V. The term with argument 2^ + 2tar', period 10,000 years, due to the figure of the Earth, has a coefficient less than o" - oi (according to Dr Hill,* o' ;, oo2 5). 399. (G) There are a few second-order terms in the motions of the planets which, when inserted in the disturbing function for the direct action, might produce sensible "■ third-order terms in the Moon's motion. A list of these was made and the possible • combinations considered, but nothing sensible was found. 400. (II) Secular variations.— It is a fact well known in the planetary theory that the secular variations of the planetary elements do not produce secular variations in the function which is the inverse distance between the planets. But the disturbing function which has-been used for the direct action depends mainly on derivatives of the planetary disturbing function with respect to the Earth's distance and mean longitude. Hence the secular motions of the planetary perihelia and nodes can produce no secular changes in the Moon's motions through the direct action. In the same way the part arising through fa! in the indirect action is insensible. We have then the ellipticity terms to consider. Here we have taken e x , the inclination of the ecliptic (which was considered as fixed) to the equator, to be constant. ...When the motion is referred to the moving plane of reference, it might be thought that this would introduce a secular term of the second order. But the principal part hi the motion of the ecliptic only produces a periodic term of period equal to that of ; the Moon s node, and the principal perturbation produced by the figure of the Ivtitb is also a periodic term whose period is that of the Moon's node plus the pivce.sMon. The term of the second order which might be sensible is therefore one having a period which is the difference between these periods, that is, a term .independent of the lunar angles and of period equal to that of the precession; it 1* therefore quasi-secular. . But it is easier to treat it by the method of § 385, and then the theorem that the first power of its period will be the only multiplier fcmdmg to make the coefficient large comes into play. If we expand the term in wie form a + bt e + ct* + . . , the portion a + bt e is only a slight alteration to the mean {''Ugilude (an observed quantity), and the secular part ct c " will therefore have the first l-"j\\cr c the period as a divisor, and consequently may be expected to be very small. * Amer. Eph. Papers, vol. iii. p. 342 ; or Coll. Works, vol. ii. p. 318, IioVAL ASTRON. SOC, VoL. LIX. 1 9 m § m ^ ■ 9° Mr. Ernest W. Brown, Theory of the The period is somewhat altered owing to the motion of t, but the argument is unaffected, since t is only present with w s . I have computed this term in the following section, and have found o' /- i5 for the coefficient of the periodic term, and consequently, for the secular acceleration durino- historic times, o' /- oooi^ 2 , a quantity quite insensible. The theorem that periodic terms of the first and second orders independent of the lunar angles can never receive multipliers higher than the first power of the long period practically enables one to reject any possibilities of secular or quasi-secular terms arising from perturbations of the second order, whatever may be the source. 401. Summary. — -We have to find : Case (B). The indirect effect of the solar terms whose arguments are 4M — 7T + 3 V, 3J — 8M + 4T. In longitude these are, according to Newcomb,* &V = + "-266 sin(332°"3 + ng°-ot,)+ 6"-4osin (22i c, i + 20°-2t). where t c is the number of centuries from 1850-0. Case (D). The term of argument Z + 3T- 10V due to substitution of the periodic variations of the coordinates of the Earth and Venus of period 13T — 8V in the dis- turbing function for the direct action. Case (F), The insertion of the motions of the perihelia and nodes in the arguments Z+16T— 18Y, Z + 3T-10V, with the resulting changes in the divisors s 2 and the; coefficients of the terms. ; Section (iii). Calculation of the Terms. J 402. Case (B). We require §*p', and we only know o 2 F. Now, 8*p' depends maiidv on <rV/V, and for terms of very long period we have "J a " m so that, approximately, But the methods of Sect, (i), Chap. X., and of Sect, (i), Chap. XII., give approximately; here the suffix o denotes the values of M 1; if 2 from Sect, (v), Chap. X. corresponding to 6 = 0. This formula, similar to that obtained for first-order terms by Radact,! who,! however, neglects M 2 , gives for the two terms mentioned in the previous section SVj, = + o"-04 sin (152° + 1 ig°-of,) + o"'84 sin (4i°'i + 2o°-2f,) . * Tables of the Sun, p. 19. f L.c, § 279, p. 39. Motion of the Moon, 91 r -i j 1 ■ IBBl • o'. Case (D). The term of argument 1 + 16T- 18V+ i5i°T> = <p arises from a of this argument in R P . The term of argument I + 3T - 10V + «° = <p - ft arises fiom a term 111 ifR. (f^ T ^ •here «'T is the term of argument ft in the Earth's mean longitude, and SV a term of rh<> same tii'ii'ument in the mean longitude of Venus. From Niavcomb's computations* SV — — (1*92/1 '44)iST, so that C T_SV= + 3 -i 6 i"-89sm(i3T-8V + i34°)= +4""4i sm (13T - 8V+ 134°) . 1 '44 T«) deduce the new term from that with argument cp which has a coefficient in SV of •14"-;. we must multiply by - 16 to account for d/dJ + d/dv,', by (13-01/1 -85)* for the tJiatisre iu the -value of s' 2 , by ST - SY, by \ for the change of the product of two smes tie cosine of the difference of dp, ft, and divide by 206265 for the reduction to seconds. This gives = -o"-i2sin(Z + 3T-ioV+i7°). Tlii- rtdu.-es the term of this period (see § 316) to + o ;/l 2 3 sin (I + 3T - 10V + 2 1°). 404. Ca„e (F)., The principal part of the term with argument 1+ 16T- 18V+ i5i° - o " mod iVom 1+ 16T- 18V + 2I1" ', where h" is the longitude of the node of Yenus. ode has a daily motion of -o"-05, so that the daily motion of the argument is ii instead of - i3"'Oi. The coefficient (i^'^o,) must therefore be diminished jviio of the squares of these two numbers, that is. by o w- 2 2. This is the only :• effect that terms of the second order have on the great Venus inequality, u- "oeffjeient of the other term depends chiefly on 1 + %T~ 1 oV + rar' + 6/&". The x.Hon of the two latter terms of this argument, which were the parts previously ••<;,. is -o"-26, and of the former three, + i"'85. Hence the coefficient must be "1 i:i the ratio ( 1*85/1 -59) 2 . It was found in case (D) to be o /; -2 3 ; its final ■• rhuvfore +o' /- 3i. '-" no, ion of -r in the argument w s — r due to the motion of the ecliptic was ■d. It has a daily mean motion of +o // '09 J so that the coefficients must be •l-'Ui by about 1/1800 of their value. 'f.. J special term in the secular acceleration due to the motion of the ecliptic 1 ?hr,',\- of the Earth.- — The arguments of the principal terms due to the motion <••hj_.de and the figure of the Earth are respectively (p- w s + g6°-2 -o"'Ggt, ■r--' m \dl, where the coefficients of t are the daily motions of t and of the ion respectively. The argument <p-ft is therefore -~o' /, 23£ + 96° - 2. This is ecnlar, but it is more easily treated as periodic by the formulas of case (A) of the * Am. JSph. Papers, vol. iii. pp. 476, 488. ||p|||H|ii liiiSlii HH! mmmk lillS '~^^^*****la*''l**mmmmmmmmmmmmmmmmmmnmm 92 Mr. Ernest W. Brown, Theory of the previous section. It is independent of the w L , and therefore only receives the first power of the period as a large multiplier. We have j\ =j 2 =j\' =>/ = o, j a =j 3 = 1 , p = b 3 - o ;/ -09, p' = b 3 + o"- 1 4 . Since b s has a daily motion of- 190" "8, we can put p=p r = b 3 , except when they occur in the combination p—p f . Substituting in the formulae of § 385, and remembering that we only need §hv u -F{ _ x 1 ( y dX -,d\'\ 1 db 3 _ XX' 1 _ db. 3 _ db s _ — — } - . .1 A -■ + A - - , ' ' ' (p-p') 2 ' P~p\ de 3 dc 3 Jbg 2 dc l p-p bJ dc-^ de a Now A, X' contain the factor k, and c s = - 2k 2 na 2 approximately. Hence ~ 1 Y V _ XX' 1 db t b., d<\ (p-p') 2 2b 3 '(p-~p')c, i di\ The second term in the square bracket is of order k 2 compared with the first, and may therefore be neglected. Treating (x/ in the same manner, and neglecting the terms factored by db 3 jdc 3 , which are of order k 2 compared with the others, we find, after some reductions, p-p " c 3 b 3 (p-p)\ dcj dcj/ c 3 b 3 (p-p) de 1 Also, H x is of order k 2 compared with these terms, Hence sin (<f> - cj>') , XX' db„ d ,,,,.. — . — s + — (XX ) h dc i dc i 1 2 (p-v) h c i Now, by Sect, (ii), Chap. X., approximately, ! i d 1 d n db , „ i = _ /3='33. ^=+-0037, & 3 =--oo 4 o, : ; dc t a~p an dit i Also, X = ka 2 nM, A' = k«V, where m, m' are independent of the c t for the terms of:; lowest order, so that AA ; = IC'ahv'p.p.' = c 3 n 2 pi", where //' is defined like p., m'. Hence i mv, = l ■ JP-, j^^lz - 6] sin (* - 4>') ■ \ But, approximately, | - 4na 2 y8y = 8c 8 = — - cos tj> , - qncpyh'y = — cos <f>' ; | ■> ■' , 1 and $y= + o"-jo cos <£ (§ 370), S'y = -4 //- 35 cos <£' (§ 374), p = b 3 = - i90 /; '8, p—p'=—o"'2T,, (p — cp' = g6''2—o /f '2^t. Hence 1 SVj, = - 12 ■ i2 — ■ 1 7 A4 — 35.; g ; n (06° '2 -o" , 2*f) = o"-i< sin(o6°-2 - 2°-%t c ) , I. - '23 206265 I which is equivalent to a modern secular acceleration of o //- oooi^ 2 , and therefore entirely insensible. 1 406. It might still be thought that terms of the second order in the disturbing function of Sect, (iv), Chap. XII., will give rise to sensible secular terms. But 9 S is 01 the third order, and therefore any terms which arise will either depend on the tv { or will be constants. It is also to be remembered that arguments in x, y containing odd Motion of the Moon-. 93 multiples of w s , and arguments in z containing even multiples of w s , only arise through the disturbing functions SR, S Z R, and much the largest of the combinations with a long ' period and of the second order is that just computed. Section (iv). Perturbations with Unknown Constants. 407- The attractions of the minor planets.— No one of these is large enough to produce any sensible effect on the motion of the Moon, but their aggregate mass may possibly be comparable with the mass of the Earth. The chief force -would be that of a riu°" of matter of diameter between two and three times that of the Earth's orbit, and the principal effect would be constant additions to the mean motions of the perigee and node of the Moon's orbit. But on any supposition which would fit in with the small differences between theory and observation for the motions of the planets, these constant addition,-: cannot exceed two or three hundredths of a second of arc in the annual mean motions of the Moon's node and perigee. Periodic effects would be smaller, and the chief of them would have a period of one vear, with a coefficient less than one-thousandth of a second of arc. 408. Other matter in the solar system.— There is undoubtedly a large number of meteoric bodies distributed through the solar system and revolving mainly round the Sun. The most reasonable supposition is that this matter may be considered as arranged in rings of varying density round the Sun as centre, so that the effect would ■ be that of a thin plate with its centre at the Sun, and of density increasing towards the centre. If this density varied as the qth power of the distance from the Sun, where q h some negative number, the effect would again be to add to the apparent mass of the Bun, to add something to the mean motions of the perigee and node, and to produce additions to the known periodic terms. The effect on the Earth would be of a similar nature. No secular terms arise. Professor Kewcomb has discussed such hypotheses.* In auv case, the effect on the Moon can be neglected. 409. The action of the tides.— First, neglecting friction, the action of the lunar tide chiefly produces a standing wave with reference to the line joining the Earth and • Moon. Its effect is therefore similar to that of the figure of the Moon, and there can be little doubt that it is quite negligible. The action of the solar tide must chiefly ..produce a term depending on the difference of longitudes of the Sun and Moon, and is similarly too small to be considered. The reaction of tidal friction produces. a real secular retardation of the Moon's motion, as well as the apparent acceleration due to the slowing down of the Earth's rotation. The former is nearly equal to the latter, and •t«e real retardation would be between two and three times the observed acceleration. I lier<- being no data on which to base any exact numerical estimates of either of these •'{Uaiitititb, the secular acceleration will be considered as an observed quantity, the magnitude of the apparent value being not very different from that (5"'8) found from the attractions of the planets. * Astronomical Constants, chap. vi. 94 Mr, Ernest W. Brown, Theory of the CHAPTER XV. THK FINAL EXPRESSION FOE THE MOON'S COORDINATES. 410. In this concluding chapter I gather together all the perturbations which have been found in detail in Chaps. X.-XIV., so that the expressions for the coordinates of the Moon in terms of the time are obtained by adding the results given below to those at the end of Chap. IX. 411. The values of the mean motions of the perigee and node are collected in the following scheme with the references. Annual mean motion of the Perigee, Node. Principal solar action (§ 195) Mass of the Earth (§§ 381, 382) Direct planetary action (§ 316) Indirect „ ,, (§ 370) Figure of the Earth (§ 374) „ ,, Moon (§ 379) Final values (epoch 1850-0) [The small differences from the values given in 1904* are chiefly due to the some-: what doubtful parts depending on the figure of the Moon ; these were neglected in the:: earlier paper. The value of the ellipticity of the Earth adopted here is that correspond-:; ing to the result marked (/3) in the paper.] I 412. Notation. — I now use w v w 2 , iv s to represent the mean longitude and mean longitudes of the perigee and node with the motions just given ; w, the mi u) longitude of the perihelion of the Earth's orbit at epoch; Q, V, T, M, J, S, the me: 11 longitudes of the planets. These will receive additions in § 413 below, denoted .1)* the symbol S. Thus D = w 1 4-8?9 1 -T-8T, l = w 1 + Sw 1 -w 2 -Sw 2 , F = w 1 + Sw 1 -w s -Bw a , I' = T + ST - vf - Bm'. f The constants y, e' also receive additions §y, <V, given below-. The changes in the Moon's coordinates are accounted for if we multiply the terms containing the arguments iF, il' by the variable factors + 146426-92 - 69672*04 •68 + '19 + 2-69 - 1 '42 •16 + - °5 + 6*41 - 6 '00 + -03 •14 + 146435-21 -69679-36 rs , . . 1 Sy 1 . Se I + . I i -i • - -—r- , I +;*,—■ , y 206265 e ' Monthly Notices, vol. lxiv. p. 532. Motion of the Moon. 95 pe>'.th>-lv. A.11 other variations of the constants present in the coefficients are be 1 ' m-cnsible or have been included in the expressions for the coordinates. The number of centuries from i850'o is represented by t e . The arrangement of the tables is the same as that of Chaps. XL, XII. All final coefficients below o' /- oo3 have been dropped. A star instead of a number in the last place denotes that the last figure has not l .ji computed. mmm 96 Mr, Ernest W. Brown, Theory of the 413. Terms added to the Arguments and to the Constants. <5-«, = + 5"-8^, 2 + 2o"'ooi C sin \\> ; + c * i3T-8V + 3 i 3 c -9 237 2l) ~l + 2lT - 20V + 2 73°'0 126 Q-4T + 239 3 2D-/ + 8T- 12V + 303 33 8M-4T+310 3 2F - 2D + 6T- 5V + 270" 54 9M- 5 T + 3 o5° 8 3Z-2l) + 2 4 (T-V) 10 HM-6T + 335" 6 D+ 12T- 15V+262 13 13M-7T+19 6 D + 25T- 23V+ 190 13 i S M-8T + 43* 26 F + 24T- 23V + 285 3 17M-9T + 63 4 D+Z-F+I7T-I8V+75" 8 ii9° - oif ( .+ 152° 4* 2D-Z+ST-4Q + U3 3 20°'2t c + 4l° m I 84* 2D-Z + T- 3 Q + io5° 75 /4-3T- ioV-2°-6f c + 3 3° 31* 2F-Z4-3T-4Q + 67 3 Z+16T-18V- i°-ot+isi°-o 1427* 4D-3Z+25M-23T + 67 4* 1+ 29T - 26V + II2°'0 108 D-F+2M4-16S 17 Z+2i(T-V) 30 n' s + 2 76 0, o 282 H' 3 + l°'4t c 7317 8w 2 = -38"-3Z c 2 -o"-ii8sin(Z+i6T- i8V+isi°-o) 4-o"'84o sin (w s + 2 76 0, 2) - 2"'092 sin (w 3 + i°'4t c ) + the ten periodic terms in Sw t whose angles are independent of w u iv 2 , w s ; s "'s= +6"'sZ, 2 + o"'i72 sin (<?+ 16T- 18V + i5i°-o)+ i"-86 sin (w 3 + 29o°-i) + i5"-58sm(t» s -o°-9Z c + 276°-2) + 96"-69sin (» 3 + i 0, 4^); ST= + i"-89 sin (13T- 8V+ i34°) + o-" 2 o sin (15M - 8T+ 216 ) - 6"'40 sin (2o°-2Z c + 4i°) -o"'2 7 sin (ii9°-o^4- 152°); 8sr' = o° , 323i ( ., 8J= +o°-33sin(38°'54+n5°). 3S = - o°-8 3 sin (38°- 5^+ 115°); 8y= - o" - o83 00s (?» 3 + 290°" 1) - o" - 698 cos (w> 3 - o°'9Z c + 2 76° '2) -4" - 35 x cos(m' 3 4-i 0, 4Q; 8e' = - - 00248e%. . ■ Motion of the Moon. m 1 m 1 P ■ i 414. True longitude = 2o /; '*oor C sin {0 +/T + i(T - V) + a°}. HI # = 179 3° 7 3S9'3 42 46 33 24 272-9 271-7 2D 34° 176 4 so 199*0 92 204 26 1 7 9 207 4 114 26 > = 2 D = Z *' a 3 * a . ' ■ 6 O II _ 7 180 4 5 O II - 6 180 5 4 O 8 - S 180 6 3 180 34 - 4 180 6 2 3 6 ~~ 3 180 8 1 l8o 23 - 2 O'O 61 1 - 99 - 1 180*0 129 2 1 79*5 136 1 o - o 152 3 178 1 3 2 iSo'o 48 4 180 4 3 180*0 127 18 3 4 180 11 1 2^2 3 1 - 2 258 4 2 271-0 40 1 75 8 3 271-5 37 2 271 46 4 89 \ 3 272 40 20 2 73 3 4 272 5 II 78 3 S 92 4 10 78 4 3 3 2 t 2- 6 9 78 4 - 1 .- 3 268 3* 8 7.0 4 . - 2 264 46 7 78 S - 1 104 9 6 78 s 1 102 3 5 84 7 2 282 7 4 78 7 3 280 7 2 271 19 2 -18 209 3 1 98 9 3 210 14 2 281 13 4 205 4 3 281 3 5 J 9 3 3 199 11 6 198 16 4 202 3 2 - S 161 3 s SO 4 - 4 33 6 4 6 162 6 - 3 33i *s 3 342 "5 3 S "5 4 2 ■ 7 3 "3, - s 65 4 15 151 4 18 151 10 Stlfe ?RON, SOC, V LIX, 13 9 8 Mr. Ernest W, Brown, Theory of the True longitude = 2o"'oo i C sin {0 +j"T + /(T - V) + a} . 0^ 2 D-/ e = 2 D-Z <9 =22 -2D J '<■ a J i a C 3 i a c O - IO 180 3 -3 340 24 4 3 - 9 i So 6 — 2 6 3 1 4 92 4 - 8 180 8 x s 151 25 _ 1 -3 268 3 - 7 180 15 3 s 114 4 -2 268 3 _ 6 180 22 2 6 17-4 62 - 5 180 3« 6 = 2 T> + 1 - 4 i8o'o 83 -5 3 i9=2l~4D - 3 i8o'o 658 -3 4 3 8 .... 2 CO 137 - 2 4 _ J 180 J 3 1 11 6 = 3/- 2D I O'O i33 2 180 iS 3 180 3 2 179-6 iS7 3 180 3 3 178 14 1 2 271 5 6= -4D 4 180 3 3 27 1 4 3 7 18 I ~~ 2 259 1 1 3 - 1 - 2 269 3 (9= -D — I 2 3 270 271-4 271-9 3 65 49 8~- 1 = 4D - Z 180 6 1 3 273 6 = 7^D 5 4 90 5 2 7 1 3 273 11 i - 8 - 7 269 269 3 S e=2i = w n - 6 269 8 — 2 S 2 3 216 19 - 5 269 24 - 1 180 10 — 2 - s 255 - 3 - 4 89 3° 1 10 ~~4 255 9 - 2 268 38 2 180 6 ~3 75 16 I IOI 9 3 1 8c 9 - 2 75 * 5 2 281 1.3 1 2 273 3 1/ 2 S3 3 - 1 - 2 267 3 (9 = w 3 + 2F 2 3 200 19 2 3 216 4' 4 202 4 <9=2?-~ 2 D - 2 -3 4o 4; 5 19 4 - 2 7 2 - 7 340 3 - I 180 S 6 = ?» 3 ± I - 6 162*6 79 2 180 3 2 5 216 3: ^ 5 165 4 3 i8o - o 73 Motion of the Moon. 99 415, True longitude = 2o"-coi C sin {0 +j"M + i(M - T) + a }. i a C I 180 11 2 l8o'2 195 5 3S7 14 4 349 s 3 260 6 ,-i 224*4 3 2 7 2 2I2'4 38 3 212-5 48 .4 33 1 10 ; 244-8 93 s 245 20 4 244 14 : -¥ 62 6 ? 2?7 16 ?4 276 13 5 275 6 6 94 #=2D 3 •:2.. S I 180 4 P' 181 44 3 s ;'i 224 23 ;;2- 212 6 ;;3 : 214 8 4 37 3 W: 149 3 |:S 328 3 1 3 1 ? 23 280 3 2 244 5 :••! ' 244 4 ;:4'- 246 4 - 297 8 f? * a C ~3 180 3 — 2 38 - 1 4 1 180 5 2 180 43 3 3 4 180 3 1 1 223*3 73 2 212 10 3 213 13 5 210 9 1 -3 33° 9 - 2 S 2 ? 8 _ 1 306-3 74 2 2 245 17 3 245 S 4 244 3 6 63 6 2 -4 296 3 -3 295 5 - 2 295 18 3 3 277 3 4 276 3 3 -4 264 3 )=2~D-l 5 180 3 4 182 20 3 5 2 1 3 1 3 1 180 8 2 iSro 61 3 3S3 • 5 -■2l)~~l i a I 220 31 2 212 II 3 214 14 4 27 3 -6 149 3 -5 I S I 43 ~4 329 3 -3 327 3 - 2 328 6 - 1 320 35 3 280 4 2 244 11 3 Z44 6 4 245 S ~6 298 33 -3 296 3 - 2 297 14 2 1 1 1 - 1 ^= 2 D4-^ 180 82 93 0=2l - 2 O 2 180 I I 232 I - I 308 :2/~2D o 209 244 4 17 18 ■ iim mm 1 IOO Mr. Eknest W. Brown, Theory of the 4 1 6, True longitude = 2o" -oo i C sin { & +j"J + i(J - T) + a' } , 2.D- l 'i a I 178-8 643 2 3S9 - 6 187 3 7 10 -3 257 6 - 2 274 18 O 289*9 87 I 241-5 .165 2 352-0 5 2 3 355 4 - i 250 10 o 324 5 i 238 2 5 2 344 6 I 230 #= 2 D 3 "3 4 — 2 i8o'o 70 - I 1 33 I I78-5 167 2 359' 2 85 3 13 7 o 349 27 i 237 35 2 35 2 x 5 - 2 8 30 - I 303 6 O 184 33 2 273 9 3 102 6 i 236 5 2 345 3 O 200 3 I no 6 i a /' i a - 2 180 36 - 1 - 1 296 18 - I I'O 144 I74-2 60 I I79-0 158 1 2 273 16 2 iSo'o 190 3 I02 7 3 21 5 1 2 1 237 6 2 274 6 2 344 3 282-3 62 : " 2 - 2 19 5 1 242 39 _ ! 291 3 2 352-5 96 2 188 7 = = 2l)+l 1 298 35 - 2 180 3 257-2 63 - I I 5 2 273 6 I 178 21 3 286 8 2 359 7 326 7 1 353 4 1 238 5 I 237 5 2 343 4 2 35 2 3 1 302 5 - 1 182 4: 2.14 7 2 2 73 i = 2D - / = 4l)-Z 4 180 4 — 2 180 r\ 3 182 22 1 358 9 ;: 2 180-3 "37 2 179 , 5j 1 1 5' - I - 2 7 3:: 1 178-4 211 2 359-2 8 9 #=2l 3 14 6 — 2 180 3 i 3 261 5 — I 2 II 1 2 310 13 I 178 12 1 5 - S 56 2 180 io| 1 237-0 46 1 O 2 93 s| 2 35 2 20 I 2 39 3 3 187 6 — I — I 301 3 1 2 7'5 436 O 247 5;| Motion of the Moon. 101 True longitude = %o"-ooiC sin {e+j"J + i(J -T) + a}. 2I-2D I .« 180 s 2 II 3 179-9 240 I72-S 284 163 3 <?= 2 Z-4D % a 2 I 2 173 # = 3?-2D 2 l8o I 2 172 c 9 7 o o 45 168 417. True longitude = 2o"-ooi C sin {0 +/'S + i'(S - T) + a'}. . a 179-6 42 8 273 21 257 . 13 297 3 D c 180 10 5 270 4 257 3 25s 4 0=z i a c - I 6 1 l80 10 2 180 3 263 12 I 257 3 — 1 283 3 277 12 :2D-l i a c — 2 l80 19 I 180 14 2 O 4 271 6 I 257 3 — 2 271 5 267 6 :¥?*S3 :. 4 1 8. True longitude — 2o"-oo 1 C sin 6. 2D 2D -I 2F-Z r + 180 9 c 10 2l'+222>'' 4 39 I + 1' + 180° 6 4 I -I' +180° 6 35 2W 3 +2°-8t+i8o° 38 s : ss;i 102 Mr. Ernest W, Bbown, Theory of the 4 1 9. Latitude = 2o" -oo 1 C sin ifi +JT + i(T - V) + « c = ± F -7 -6 ~5 -4 -3 - 2 - 1 1 2 2 3 •s ■ 2 3 •3 1 o 2 180 2 2 73 (9 = F + 2l) 2 O 1 o 2 l8o ?= -F + 2D l8o ISO l8o l80 O o 180 o 180 271 272 270*0 269 199 341 6= ±F + ? o 180 o 180 180 C 9 4 6 3 5 9 23 45 21 5 16 23 9 6 68 6 3 S 6 = -F + 2D-Z i a c -4 180 4 -3 180 29 - 2 6 - 1 5 1 6 2 180 8 I 2 271 3 2 -6 162 4 = -F+2D-/ -4 180 4 -3 t8o 3i — 2 6 — 1 5 1 6 2 180 8 1 2 271 3 2 -6 162 4 i9 = F-2D + ! 1 S 9° = F-2D-Z 1 5 270 0= ±F + 2Z-2D 180 18 -6 -5 -4 -3 - 2 - 1 i 2 3 3 -7 -6 """' 5 -4 -3 - 2 - 1 1 2 3 5 -5 285 285 285 285 285 285 105 io 5 105 2156 2 55 2 55 2 55 255 51-6 75 75 75 7 5 75 125 67 3 3 6 = w 1 + 1 3 216 3 75 3 3 6 3 2 55 3 6 9 1.4 •■' . : 7'>i M,,ii,iy, ,,/ the M'iOiK [() I /=F-2l) i-- ± V I- / - j L) '■ j ; ilfliltlsf a c O - 2 fiSlllflllll? 8 2 iSo 3 I iiiilif --3 5 mmwa llllllll I - 1 3 !.6 "1 l*S 1 .345 421 . l.'Hihuh- -- ^.i"-mi C -in ','■' + /'"■) + /(.I -T) + >i' j. . - I' ii I'-:jI> 1 I80 o 37 f.'.- -F-K'P 3 '- s '' ! I So 2 O - >5° i -37 c i - 1 - -7.? 5 2<> <> -'I 12 7 fi 4 a. iSo 180 o " + F + / 1 1 100 2 i So O 3°' -' 353 -4° 422. L<<t'ihi(h ! — ^o""no! (.' sili v'/. - 1" - 2 1 • ?'-! :■ 1S0 -'■. ■ - T + 7 5' 42.;. /'.'/■-///,/.,■ ^;. 'voi < ' i-ii< w 2 1 ) / ,v t v . 1 <o : ,|»-/-2i.I -T)-i.So" .'.I.) ■ / - 3..I H- 2T + 7' 3 ^ + |.' + /..;]l I. so O O 17- H- il:'+:/-:D c iliilfiiiiiiiii c* 5 w i + <r. + 2 -s, ■\? 7 MJ, ~ w 8 - _T 3 S "•1 + w 8 illlilllllli 5 7 •' r i - / -i- 1 No - IbO 172 34 10.3 C 3 c 4 1 1 5' 20 7 35 ai ■y-u WMKm wSjffi III HI m» ■■■ ill •i »:•»:■;; ■I flHI flfr lis. itr «■ IBE. ■ ■ IMP™ i - : : : : :: :':-: fl «■■■ ■■111 !■■■ '■: : &- JHBHi • .- ■■■.-.• . ; 1.' l'Niw-.i:-ir-Y, 1 - - s -1/"'''/ 3. FlVIH XH